diffusion of aromatic solutes in aliphatic polymers above glass...
TRANSCRIPT
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Diffusion of aromatic solutes in aliphatic polymers above glass 1
transition temperature 2
Xiaoyi Fang2, Sandra Domenek
2, Violette Ducruet
1, Matthieu Refregiers
3, Olivier Vitrac
1* 3
1INRA, UMR 1145 Ingénierie Procédés Aliments, F-91300 Massy, France 4
2AgroParisTech, UMR 1145 Ingénierie Procédés Aliments, F-91300 Massy, France 5
3Synchrotron SOLEIL, l’Orme des Merisiers, F-91192 Gif-sur-Yvette, France 6
Abstract 7
The paper presents a harmonized description of the diffusion of solutes with repeated aromatic 8
jumping units (JU) in entangled aliphatic polymers above their Tg. It is shown that the trace 9
diffusion coefficients, D, are scaled with the number of jumping units or equivalently with solute 10
molecular mass, M, as 1 K T Tg KM M
, where Kα and Kβ are temperature-equivalent 11
parameters related to Williams-Landel-Ferry (WLF) ones. Kα is almost a generic constant for 12
aliphatic polymers. The scaling of diffusion behaviors of linear aliphatic and aromatic solutes 13
appear separated by a temperature shift, Kβ, of ca. 91 K. The effects of the number of JU and the 14
distance between two JU were specifically probed in several aliphatic polymers (polypropylene, 15
* Author to whom correspondence should be addressed.
E-mail: [email protected]
Tel. +33 (0)169935063
mailto:[email protected]
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polylactide and polycaprolactone) at different temperatures above Tg with two homologous 16
solute series: short oligophenyls and diphenyl alkanes. An extended free volume theory for many 17
JU was accordingly inferred to account for the observed statistical independence between the 18
fluctuations of the free volumes probed by each JU and the probability of the collective 19
displacement of the center-of-mass of the solute. Outstanding properties of short oligophenyls 20
series provided further insight on the underlying molecular mechanism of translation. Their 21
activation energies grow differently according to the number of phenyl rings, NPh, being odd or 22
even. Constrained molecular dynamics demonstrated that such a parity effect could be 23
remarkably reproduced when the translation of each JU (i.e. phenyl ring) was randomly 24
controlled by a combination of short and long-lived contacts. 25
Keywords: diffusion, aromatic molecules, aliphatic polymers, scaling exponents, free volume, 26
activation energy, molecular dynamics 27
28
1 INTRODUCTION 29
No general diffusion model is available to predict the broad range of trace diffusion 30
coefficients (D) of organic solutes such as oligomers, additives, residues, contaminants or 31
degradation products in polymer materials at solid state, which corresponds to their conditions of 32
service. By analyzing D values in polyolefins, a strong dependence of D values has been 33
highlighted for several categories of organic solutes: resembling the polymer (e.g. linear or 34
branched solutes) or not (e.g. aromatic molecules, hindered antioxidants)1. The exponents, α, 35
scaling D with molecular mass (M) as D∝M-α, were found strictly greater than 1 and typically 36
above 2, which reflects variations of D over several decades with only small change of M, 37
between 102 and 10
3 g⋅mol-1 for most of the molecules of technological interest such as 38
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antioxidants, UV stabilizers, plasticizers,… Those variations were related mainly to three 39
geometric factors: molecular volume, shape factor and gyration radius. D values reported by 40
Berens2 suggest that mass/volume dependence is even greater when below the polymer glass 41
transition temperature (Tg). On the opposite, in melts, scaling exponents for homologous alkane 42
series in polyethylene were experimentally assessed close to 1 in agreement with the Rouse 43
theory3. Molecular dynamics simulations of generic polydisperse systems above Tg found also 44
exponents of 1 for a wide range of thermodynamical conditions4. The absence of temperature 45
effect on α was, however, not verified experimentally. Thus, Kwan et al.5 reported a strong effect 46
of temperature on α values – with α falling from 4.7 to 2.1 between 23°C and 85°C – for n-47
alkanes dispersed in a lightly cross-linked amorphous polyamide, suggesting a continuous but 48
sharp evolution from Tg to higher temperatures and without a significant contribution of the 49
possible crystalline phase. 50
Such a high mass dependence, with α values varying between 26 and 3
7 have also been 51
reported for the self-diffusion of entangled polymer chains subjected to reptation and strong 52
reptation translation mechanisms respectively, with intermediate values close to 2.4 when 53
reptation is combined with a constraint release mechanism8. The formal analogy is, nevertheless, 54
of a limited use for solutes with molecular masses ranging between 102 and 10
3g⋅mol-1, as their 55
gyration radii are much smaller than the typical entanglement length of polymer segments. For 56
organic solutes larger than voids between polymer segments and smaller than entanglement 57
length, only a partial and local coupling between the reorientation and local relaxation modes of 58
the polymer can be expected9. It was thus shown that aromatic molecules remain non-oriented in 59
the amorphous phase of polyethylene when the material was stretched uniaxially10, 11
, whereas 60
anisotropic diffusion of toluene was observed in compressed natural rubber12
. 61
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The main goal of this study is to provide a consistent description of molecular diffusion 62
mechanisms of aromatic solutes mimicking molecules of technological interest in linear 63
polymers and to provide a polymer-independent description of the strong dependence of D with 64
M at any temperature greater than Tg. In a first approximation, such solutes can be described as 65
the repetition of rigid jumping units subjected to independent displacements on short time scales. 66
An extended version of free volume theory is accordingly introduced to account for the weak 67
coupling between polymer relaxation, controlling the fast displacements of each jumping unit, 68
and the collective reorientation and translation of the whole solute. It was validated on two 69
independent homologous series of solutes. The first series of diphenyl alkanes with close size 70
was used to probe polymer effects (i.e. independent motions of jumping units) at a similar 71
reference temperature. The second series, short linear oligophenyls, probed specifically solute-72
related effects (i.e. the collective motions of repeated jumping units) and their activation by 73
temperature. As both series started with a common first molecule, biphenyl, entropic effects 74
could be reliably scaled for both series. 75
This paper is organized as follows. Scaling laws for linear aliphatic and aromatic solutes in 76
aliphatic polymers are discussed in section two in the framework of a coarse-grained theory of 77
diffusion of solutes consisting in a linearly repeated jumping units or blobs. The existence of at 78
least two correlation time-scales between the displacements of jumping units and surrounding 79
host particles is used to justify the major deviation to the Rouse theory13
and to conventional free 80
volume theories in dense entangled aliphatic polymers (i.e. far below their melting points). An 81
extended free volume theory is proposed based on a formal separation of thermal expansion 82
effects acting on the displacements of each jumping unit, so called “host effect”, and barrier 83
effects, so called “guest effects”. A simple theory of activation barriers for the diffusion of 84
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oligophenyls is accordingly proposed. Section three reports the methodologies used to determine 85
trace diffusion coefficients, ranged between 10-17
m2⋅s-1 and 10-12 m2⋅s-1, in three different 86
polymers: polypropylene (PP), polycaprolactone (PCL) and polylactide (PLA) for both 87
homologous solute series. A fourth polymer, plasticized and unplasticized polyvinyl alcohol 88
(PVA), was used as external validation for arbitrary polymers above Tg. The proposed free 89
volume theory is tested against experimental in section four. The scaling of activation energies 90
and entropies of oligophenyls is discussed according to results obtained in constrained molecular 91
dynamics simulation. The likely mechanism of translation of aromatic molecules in linear 92
polymers above Tg and its consequence on solute mass dependence is finally proposed in the last 93
section. 94
95
2 THEORY 96
This section discusses the possible causes responsible for the deviation of solutes larger than 97
voids to existing diffusion theories in linear polymers at solid state. By relying on the diffusion 98
properties of solutes including linearly repeated jumping units (or also called blobs), two major 99
arguments are proposed: i) the increase in polymer density with decreasing temperature down to 100
Tg affects the individual displacements of each jumping unit, ii) and the resulting displacement 101
of the center-of-mass is strongly affected by the heterogeneous dynamics of individual jumping 102
unit. 103
104
2.1 Scaling of D with the number of jumping units and temperature for linear solutes 105
The strong slowdown of D values with M (respectively the number of jumping units in solutes 106
with linearly repeated patterns) is major characteristic of organic solutes in polymers at solid 107
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state (e.g. semi-crystalline polymers or polymers below their flow thresholds). Published values 108
are scarce, mainly due to the necessity to work with homologous structure series and a wide 109
range of D values. Scaling of D in polyethylene far above its Tg is reported in Fig. 1 for two 110
solute series artificially split into two categories : i) linear aliphatic solutes (Fig. 1a), including 111
linear n-alkanes (n=12..60)3,
14
, n-alcohols (n=7..18)15
and esterified phenols containing a 3(3,5-112
di-tert-butyl-4-hydroxyphenyl) head and a long n-alkyl tail (n=6..18)16
, and ii) aromatic solutes 113
(Fig. 1b), including n-alkylbenzene containing short n-alkyl chains (n=0..4)17
, substituted and 114
hindered phenols (M= 94-545 g⋅mol-1)15, respectively. For each series, D values are scaled as a 115
power law of M (ranging between 70 and 103 g⋅mol-1), whose exponents are much greater than 116
unity and tend to decrease when temperature is increasing. α values assessed up to 4 or 5 117
constitute a major deviation to the Rouse theory that predicts values close to unity instead. 118
According to the Rouse theory, unitary values are related to a purely single chain relaxation with 119
a uniform friction factor and without any chain end effects13
. Mutual diffusion coefficients of 120
n-alkanes in polyethylene melts3 extrapolated to infinite dilution suggest that α values close to 121
unity should be recovered far from Tg. In agreement with experimental results and free volume 122
theories, we propose a scaling exponent deviation to the Rouse theory, denoted Δα, which 123
depends mainly on the temperature difference T-Tg: 124
1T Tg T TgM MD M
(1) 125
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126
Figure 1. Log-log plots of trace diffusion coefficients in polyethylene (PE) either with low 127
density (LDPE) or high density (HDPE) of a) linear aliphatic and b) aromatic solutes. Dref was 128
chosen as the diffusion coefficient of the first solute in the considered solute series. 129
130
For solutes consisting in a small number of linearly repeated jumping units or blobs, N, the 131
deviation to the Rouse theory close to Tg can be thought as the consequence of irregular jumping 132
unit displacements with time. To justify this argument, we adopt the coarse-grained model of 133
Herman et al.18-21
initially proposed to describe the dynamics of flexible linear chains in the melt. 134
Similarly, by neglecting end-effects, we assume that the mean-square-displacement of a single 135
jumping unit increases with time as: 136
6 blobg t D t t (2) 137
where Dblob(t) is the time-dependent diffusion coefficient of a blob/jumping unit. For any solute 138
with identical jumping units indexed i=1..N, Dblob is assumed to decrease with the amount of 139
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cumulated pair correlations between the displacements of jumping unit i, denoted Δri(t), and the 140
displacements of all particles within the system (i.e. including host and solute), denoted Δrj(t), 141
as: 142
0 0 0
,
2
·blob
i j
all j including i
all j including i
D D DD t
C t C tt t
t
i j
i
Δr Δr
r
(3) 143
where D0 is a scaling constant. In this simplified description, jumping units have a similar 144
spherical shape and volume regardless the detailed chemical structure of the solute and host 145
polymer chains. An increasing C(t) with time leads to a sub-diffusive regime. The mean-square-146
displacement of the center-of-mass (CM) is accordingly given by the covariance of the averaged 147
displacements of all jumping units: 148
, ,2 21 , : , :
1 1 22
N N N
CM i j i j
i i j i i j jij
g t g t C t g t C t g tN N N
(4) 149
For a linear and flexible solute and by neglecting torsional constraints, only the N-1 150
correlations between the displacements of connected jumping units are significant. Eq. (4) 151
becomes accordingly: 152
2
12
t
CM connect
g t Ng t C g t
N N
(5) 153
wherein tconnectC is the normalized correlation between the displacements of two connected 154
jumping units. Incorporating Eqs. (2) and (3) yields finally: 155
00
61 1 1 12 1 6 t
connect
t
connectCM C C t
C D tg t D t
N C t N C t N C t
(6) 156
By assuming that t
connectC is small comparatively to correlations with the displacements of host 157
polymer segments in a dense system, the mean-square-displacement of CM and consequently the 158
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tracer diffusion coefficient, lim 6t CMD g t t , is scaled as 1/N, in agreement with the 159
Rouse theory13
. 160
161
In presence of strong coupling between the lateral displacements of solute jumping units and 162
host particles (e.g. large jumping unit), the translation of each connected jumping unit is 163
expected to be highly heterogeneous and controlled by a combination of short and long-lived 164
dynamic contacts as discussed in general terms in22
and23
. In the followings, we will assume the 165
simplest case where two relaxation modes resulting of many body interactions can be 166
independently applied to each jumping unit: C(t) and ttrappedC , with respective probabilities 1-p 167
and p. ttrappedC ≫C(t) is the total correlation when a jumping unit is almost “blocked” or 168
significantly hindered by host segments. From this simplified description, gCM is governed by the 169
superposition of all possible partitions between “trapped” (i.e. long-lived contacts with host) and 170
“untrapped” (i.e. short-lived contacts with host) jumping units. By assuming that g(t) still obey to 171
Eqs. (2) and (3), Eq. (6) is replaced by: 172
2 21
0
11
61
t
trapped
N j jumping units translate according to C t
all jumping units j jumping units translate according to Ctranslate according to C t
NN j
tj trappedCM
N N j jp p
NC t j N C t Ng
p
t C
D t
1
1
2
1
1
2 1 1 1 1
2 12 1
2 1
t
trapped
NN j
j
N
Np N
CN
N
C t
Np
j
N
N C t C tN
(7) 173
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wherein N
j
is the number of combinations to choose exactly j jumping units among N with 174
long-lived contacts (governed by ttrappedC ) and p
j is the related probability by assuming that j 175
jumping units are blocked independently. 176
Eq. (7) stresses that a higher dependence to N (α>1), and therefore to M, is expected for small 177
N (typically lower than 5) as soon as p→1 and while several populations of contacts between 178
jumping units and surrounding coexists. Considering only two populations provides a rough 179
scaling of D with the number of jumping units, but its main interest is to highlight that the 180
scaling with N or M would be the result of the heterogeneous dynamics of the coupling between 181
the fluctuations of free volume (i.e. associated to the displacements of a single jumping unit) and 182
collective displacements of many jumping units. 183
As the value p is related to the probability to find a free volume close to each jumping unit, we 184
propose to test against experimental values a phenomenological temperature superposition model 185
where the temperature shift factor Δα is a function of the reciprocal fractional free volume in the 186
polymer, written as for any T>Tg-Kβ: 187
1 1K
T Tg T TgT Tg K
(8)
188
where Kα is a scaling constant and Kβ is a constant depending on the size of the jumping unit. 189
Accordingly Kβ is expected to be larger for aromatic solutes than for aliphatic ones. It is 190
important to note that Eq. (8) is a special form of the Williams-Landel-Ferry (WLF) equation24
191
applied to the scaling relationship in Eq. (1). Indeed, by following the suggestions of Ehlich et 192
al.25
and Deppe et al.26
, D(T), relative to its value at Tg and to a reference molecular mass, M0, 193
can be written as: 194
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0 0
ln ln ( ) ln lnTD T KM T Tg M
a Tg TD Tg M T Tg K MK
(9)
195
which can be identified to the standard WLF equation 1
2
ln TC T Tg
aC T Tg
, with 196
1
0
lnK M
CMK
and 2C K . The absolute value of Kβ appears in Eq. (9) to keep the 197
consistency D(T)>D(Tg) for any T>Tg-Kβ. More rigorously, a temperature greater than Tg-Kβ 198
should be chosen instead of Tg. Besides, M0 should be chosen as the unity or as a multiple of the 199
molecular mass of one single jumping unit. 200
When applied to a broad range of temperatures, Eq. (9) implies a significant deviation of 201
activation of diffusion from Arrhenius’ law and a log-dependence of the apparent activation 202
energy, denoted Ea, with M: 203
2
2
0
ln ,, K ln
1 Tg+K
D T M RT MEa T M R
T MT
(10) 204
Similar deviations to Arrhenius behavior was already proposed by Deppe et al (see Eq. (7) in26
) 205
for aromatic solutes in rubbery poly(isobutyl methacrylate) near its Tg. An Arrhenius behavior is 206
expected to occur only when T≫Tg-Kβ. A similar log-type scaling of Ea with M is consistently 207
inferred for n-alkanes in low-density polyethylene from activation energies reported in15
, with 208
values of 57, 66 and 107 kJ⋅mol-1 for dodecane, octadecane and dotriacontane respectively. From 209
D values collected in17
, the same trend is also drawn for alkylbenzene in polyethylene. As 210
discussed in25
, the dependence of WLF parameter C1 to 0
lnM
M could be related either to a 211
stronger coupling with matrix mobility or to an increase in the volume of the solute jumping unit. 212
213
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2.2 Conventional free volume theories 214
Free volume theories argue that the translation of the center-of-mass of the solute is controlled 215
in time by the redistribution of voids around the solute due to thermal fluctuations. However, 216
early models derived from viscosity theories27
, such as the one proposed by Cohen and Turbull28
, 217
tend to underestimate the mass dependence by assuming that the whole solute translates as a 218
single jumping unit, which yields a scaling exponent close to 1. Theories modified by the 219
Vrentas and Duda model29, 30
for polymer-solvent mixtures incorporate two additional features: i) 220
an internal energy change is required to initiate a translation of the center-of-mass and ii) an 221
increase in free volume due to polymer thermal expansion. The corresponding mutual diffusion 222
coefficient of the solute indexed 1 within the polymer indexed 2 is given by Eqs (11)-(13): 223
1 1 2 2
1 0
V VD exp exp
V̂ /FH
ED
RT
(11)
224
P SE E E
(12) 225
11 21 1 12 22 2
1 2
1 2
ˆ K K Tg K K TgFH
T TV
(13) 226
where 1 and 2 are the mass fractions in species 1 (solute with glass transition temperature 227
1Tg ) and 2 (polymer with glass transition temperature 2Tg ) respectively. 0D is a constant. *E 228
is the effective energy barrier per mole to overcome attractive forces and defined as a balance 229
between the barrier in dilute state ( 1 0 ), EP, and in pure solvent ( 1 1 ), Es. *
1V and *
2V are 230
the specific hole free volume of solute and polymer required for a jump respectively. ˆFHV is 231
the effective (including overlaps) average hole free volume per unit of mass of mixture. ξ is the 232
solute fractional jumping unit. In the case of our studied linear aromatic solutes, it could be 233
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envisioned as the ratio of the solute jumping unit (i.e. phenyl ring) to the polymer jumping unit. 234
1,2i i
, 1,2ii i
K
, 1,2; ,2; 1i iij j j
K
, are parameters that account for overlapping factors, free-235
volume parameters and of their interactions respectively. 236
At diluted state ( 1 0 ), which is of technological interest for most of the additives and 237
polymer residues, Eqs. (11)-(13) can be recast in a simpler model: 238
2 21 0
12 22 2
0
2
VD exp exp
K K Tg
= D exp expTg
a
b
ED
RT T
KE
RT TK
(14) 239
Original constants ξ, γ2, K12 and K22 are recast into two lumped parameters named Ka and Kb 240
respectively to their counter parts Kα and Kβ in Eq. (8). In the remainder of the paper, we drop 241
the indices 1 and 2, and Tg2 is replaced by Tg. As Eq. (8), Eq. (14) is also related to a special 242
form of WLF equation24
, with an additional temperature-related translation term induced by the 243
energy barrier E*: 244
1 1
ln ln aTb b
D T KE T Tga
D Tg R T Tg K T Tg K
(15) 245
When E*=0, Eqs. (15) and (9) are equivalent with 0lnaK K M M and bK K . According 246
to Vrentas et al.31
, E* is related to the variation of internal energy when a solute is introduced at 247
infinite solution in the polymer. This quantity is expected to be low for any solute with good 248
solubility in the polymer at the considered temperature. 249
250
2.3 Extended free-volume models for aromatic solutes in aliphatic polymers 251
The need to extend conventional free volume theories for large solute in amorphous polymers 252
has been already discussed for plasticizers in PVC 32-36
and for flexible and semi-flexible solutes 253
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that move in a segmentwise manner in25, 26
. Indeed, translation of additive-type solutes invoke 254
relatively high activation energies, ranging typically from 50 to 150 kJ·mol-1
9, 37
. As such values 255
are close to the activation energies of polymer relaxations, it is thought that large or bulky 256
solutes involve the same cooperative motions of polymer segments as observed in viscous flow 257
38,39. Conventional free volume models do not provide any indication on the strong scaling of D 258
with the size of solutes, on the origin of temperature dependence of α with M for linearly 259
repeated solutes and on the size of the elemental jumping unit. All solute-related effects are 260
gathered in Eq. (14) into a single parameter, independent of temperature, , lumped within a 261
single parameter 2 2 12*a VK K combined with the contribution of the polymer thermal 262
expansion. According to this description, the translation of the solute center-of-mass and 263
polymer relaxation would be simultaneous and interrelated phenomena at rubbery state: one 264
translation of polymer segments causing necessarily the translation of center-of-mass of the 265
whole solute regardless the size of the solute. It is however well established that though the 266
activation volumes of translational diffusion of additive-type molecules are significant, ranging 267
from 80 to 250 Å3, they are smaller than the solute itself
40. As a result, a piece-wise translation 268
based on the translation of elemental flexible units appears more likely. Besides, the possible 269
absence of exact match between the shape of free volumes freed by the polymer and the shape of 270
elemental jumping units should promote a delay (i.e. long-lived contacts) between the effective 271
translation of the jumping unit and un-concerted motions of polymer segments. This last 272
assumption is very likely in the light of experimental results showing that aromatic solutes 273
remained non-oriented with aliphatic polymer segments10, 11
. 274
In this work, we adopt the following description. The displacements of each individual 275
jumping unit obeys to conventional free volume theories on short time scales but the collective 276
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displacement of the center-of-mass of the solute (CM) is controlled by the necessity of concerted 277
motions of all jumping units on long time-scales. As a result, the joint probability of a solute 278
translation of CM was associated to two favorable events: a favorable fluctuation of the contour 279
of polymer segments and the concerted displacements of all jumping units. As both events are 280
assumed to be independent, Eq. (14) was refactored for solutes made of repeated jumped units as 281
two independent exponential factors (i.e. associated to two independent marginal probabilities): 282
.
, ,, ,
, ,
, exp
solute polymerexcess
asoluteexcess b
D T Tg soluteD T solute D T Tg
D T Tg reference solute
KD T solute
K T Tg
(16) 283
where soluteexcess
D is a polymer-independent dimensionless factor enabling the extrapolation to solutes 284
comprising the same type of jumping units as the reference one, but with a different combination 285
or repetition. Although initial considerations are different, Eq. (16) resembles Eq. (9) used in17
to 286
describe the diffusion of n-alkylbenzene in polyethylene via a hybrid model. In our case, the 287
concept of “polymer effect” is related to the probing of free volume effects by a single jumping 288
unit as discussed in25
. For solutes based on the same jumping unit (i.e. phenyl ring in linear 289
aromatic solutes), this contribution is expected to depend only on the polymer host and not on 290
the solute. 291
292
Eqs. (2)-(7) suggests a complex dependence of D with the collective displacements of jumping 293
units. The chosen solute series enabled to test two effects: contribution of the distance between 294
two jumping units (diphenyl alkanes series) and contribution of the number of repeated jumping 295
units (oligophenyls series). 296
297
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By assuming that the probability of concerted motions between two jumping units decreases 298
exponentially with the distance between two jumping units, Eq. (16) was written for diphenyl 299
alkanes according to the number of carbons between two phenyl rings, NC, (from 0 to 2), as: 300
10
, ,exp ln10 exp
, 0
C C a
C C b
D T Tg N N K
D T N N K T Tg
(17) 301
where 10C
N is the number of carbons required to decrease D by 10. The first exponential in Eq. 302
(17) emphasizes that increasing the distance between two phenyl rings (higher NC) strengthens 303
considerably C(t) (and possibly Ctrapped). In particular, the polymer host is thought to fill the 304
space between two phenyl rings as the consequence of a reduction of the solute excluded 305
volume. As the number of phenyl rings are the same in diphenyl alkanes, Eq. (17) offered an 306
opportunity to assess with a good accuracy of polymer host effects by fitting Ka and Kb on D 307
values obtained in different polymers (i.e. with different Tg) and at a similar reference 308
temperature. 309
310
Regardless the host polymer, Eq. (10) predicts that the change in α values (assessing the 311
collective motions of jumping units) with temperature should be followed by an increase of the 312
apparent activation energy of D with N. From Eq. (5), cooperative displacements of jumping 313
units are expected significant only for small N values and if tconnectC and C(t) are of the same 314
magnitude order. Such effects were accounted by considering a solute related free energy barrier, 315
which varies with the number of phenyl rings, NPh: solute Ph solute Ph solute PhA N Ea N TS N , 316
where Easolute, and Ssolute are the solute translation activation energy and entropy respectively. In 317
the canonical ensemble, soluteA is the Helmholtz free energy associated to the probability to find 318
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a significant translation of CM when all phenyl rings are subjected to correlated displacements 319
with the surroundings. As in Eq. (17), Eq. (16) was written relatively to biphenyl as: 320
0
2, ,
exp exp2
exp
solute PhexcessPh a
bsolute Ph
A ND T Tg N K
RT K T TgA ND
RT
(18) 321
with 2 2solute Ph solute Ph solute Phexcess
A N A N A N for NPh≥2 and 2solute Phexcess
A N =0 for 322
NPh=2. 323
324
2.4 Modeling of activation terms for oligophenyl solutes 325
The assumption of solute PhA N controlled by a partitioning of the correlation with the 326
displacements of surrounding jumping units as C(t) and Ctrapped(t), with probabilities 1-p and p 327
respectively, was tested by constrained molecular dynamics for the simplest scenario: 328
Ctrapped(t)→∞ and C(t)>0. The advantage of this scenario is that it can be simulated directly 329
molecular dynamics simulation on isolated molecules by assuming that one or several rings have 330
their positions fixed. 331
Easolute and Ssolute versus NPh were calculated respectively in order to enable a comparison with 332
experimental values. Easolute was defined according to the typical temperature-dependent 333
translational time of CM, expring Ph Pu hsol teEa RN TN , to cross a distance equal to the 334
diameter of a phenyl ring ∅ring when 0 to NPh rings are blocked. As in Eq. (7), ring PhN is 335
obtained by averaging over all possible configurations to block phenyl rings: 336
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1..
1
0
1 1 1 1
1 1
1 , ,
1 1
Ph
Ph Ph
Ph Ph
Ph
Ph Ph
Ph Ph
j N rings are trapped
N Nno ring trappedj jN N
N j j
j k j k
ring Ph trappedN NN NPh Phj
P
j
j j
hp p j k p j k
NN N
p p p
N
pj j
(19) 337
where τ(j,k) is set the minimum time to induce a displacement of CM equal to ∅ring with exactly j 338
rings blocked within the kth
configuration chosen among the PhN
j
possibilities: 339
2
2
,
, mi 1n 1ring
j
P
k
hj k for jt t
N
CMΔr
(20) 340
The special cases, where no ring is blocked and where all rings are blocked, were associated to 341
τ0 and τ(NPh,1) respectively. τ(NPh,1) was expected to be large but finite, so that it can be set to a 342
constant τtrapped. Beyond τ(j,k), CM is likely to displace to a distance larger than one ring so that a 343
different combination of trapped and untrapped rings is expected to occur. 344
According to Eqs. (19) and (20), translation of CM occurs mainly as a sequence of macrostates 345
where 1..NPh-1 rings are randomly blocked. The transition from one macrostate to the next one 346
occurs at the fastest rate enabled by the dynamics of the CM when the solute is subjected to 347
topological constraints. 348
349
The translational entropy, Ssolute, which measures the number of microstates associated to a 350
macrostate was calculated analytically according to the same framework but at atomistic scale. 351
All possible displacements of atoms were described as the superposition of 3NA-6 quantum 352
harmonic oscillators, with NA the number of heavy atoms (i.e. carbons for tested molecules) in 353
the considered solute. Absolute entropies were calculated according to Eq. (21), as justified in41
: 354
-
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3 6
1
ln 1 expexp 1
A
i Bi B
i
N
conformation
B
al B
i
kS k k
k
TT
T
(21) 355
where the quasiharmonic frequencies Bi ik T were calculated from the eigenvalues, λi, of 356
the covariance matrix of the fluctuations of atom positions: 357
1 1 1 22 2,i i i i i ir r r r (22) 358
where i1 and i2 are carbon coordinate indices chosen among 1…3NA. 359
The translational entropy was calculated by the entropy difference when the position of CM 360
was fixed or not, as described in42
: 361
conformational conformational, , ,fixed CMS j k S j k S j k
(23) 362
Finally, Ssolute was obtained by averaging over all possibilities to block any combination of 363
phenyl rings in the solute: 364
0
1 1
1
1 ,
1
Ph
Ph
Ph
Ph
Ph
N
jNN j
ring
j k
solute Ph NN Ph
P
j
j
hp S p S j k
S NN
N
p pj
(24) 365
where S(NPh,1)=0 due to the absence of available degree of freedoms. S0(NPh) is the entropy 366
when no phenyl ring is blocked. 367
368
3 EXPERIMENTAL SECTION 369
3.1 Materials 370
Tables 1 and 2 list the studied aromatic solutes and polymers respectively. The two tested 371
series included phenyl rings as elemental jumping units and shared biphenyl as common 372
molecule. In details, the diphenyl alkanes series enabled to assess the effect of the distance 373
-
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between two elemental jumping units for different T-Tg values while keeping T close to a same 374
reference temperature. By contrast, the oligophenyls series was used to assess the effect of the 375
number of jumping units and the effect of collective barriers (Easolute, Ssolute) by shifting both T 376
and Tg values. 377
Table 1. List of studied aromatic solutes 378
Series Solutes
Diphenyl alkanes biphenyla Diphenylmethaneb Bibenzylb
Structure
M (g⋅mol-1) 154.2 168.2 182.3
NC= 0 1 2
Oligophenyls biphenyla p-terphenyla p-quaterphenyla
structure
M (g⋅mol-1) 154.2 230.3 306.4
NPh= 2 3 4 asupplied by Sigma-Aldrich Chemistry (Germany) with purity of 99.5 %.
bsupplied by Acros 379
Organics (France) with purity of 99 %. 380
381
Effects of T-Tg ranging from 10 K to 110 K on solute diffusion coefficients were investigated 382
in four different polymer hosts above their respective Tg, including: polylactide (PLA), 383
polypropylene (PP), polycaprolactone (PCL) and polyvinyl alcohol (PVA). Tg of PVA was 384
modulated by using it both at dry state and equilibrated at an intermediate relative humidity of 385
21% with 2.4 wt% of water content. Ka and Kb values used in Eqs. (17) and (18) were 386
exclusively fitted on D values of diphenyl alkanes obtained in PLA, PP and PCL at 343 K, 333 K 387
and 323 K respectively. PVA data were used exclusively for external validation purposes. 388
389
Table 2. Information and characterization of processed films 390
-
Page | 21
Polymer Tg (°C)
Crytallinity %
Thickness (mm) × width or diameter (mm)
Film processing
Supplier/product reference
Polylactide (PLA)a 60 23.6 0.02×600 Extrusion Treofan (Germany)/ BiophanTM
Polypropylene (PP)a 0 55.5 0.2×800 Extrusion blowing
Borealis (Austria)/ HD621CF
Polycaprolactone (PCL)b
-60 50.3 0.01-0.04×200 Solution casting
Creagif Biopolyméres (France)/ CAPA 6800
Polyvinyl alcohol (PVA)c
55 82
50.0 0.01-0.03×200 Solution casting
Sigma-Aldrich (USA)/ Mowiol® 20-98
aFilms were processed at industrial scale and used as received.
bPCL films with molecular 391
weight of 8⋅104 g⋅mol-1 were processed at laboratory scale as described in43. cMolecular weight 392
of PVA is 125000 g⋅mol-1 with 98.0-98.8 % of hydrolysis degree. Tg of PVA films were of 82°C 393
and of 55°C, at dry state and when the films were equilibrated at a relative humidity of 21% 394
respectively. All films were processed at laboratory scale as described in44
. 395
396
3.2 Film processing and formulation 397
PLA and PP were supplied as films and used as received. PCL and PVA films were processed 398
by solution casting, according to43,44
respectively. PCL dissolved in dichloromethane with 399
concentration of 2.4 wt% was poured into a glass petri dish. After 24 h of evaporation at room 400
temperature, PCL films were peeled off from the dish and dried in an oven at 30 °C for three 401
days. PVA films were formed by applying the similar process by using deionized water as 402
solvent with PVA concentration of 2 wt%. The evaporation of water took at least three days. 403
Then, the films were conditioned under controlled relative humidity. 404
Tg and crystallinity degree of each polymer were measured by differential scanning 405
calorimetry (model Q100, TA Instruments, USA) at a heating rate of 10 °C/min within 406
temperature limits adapted to each polymer. The crystallinity degree was calculated from the 407
-
Page | 22
melting endotherm in the first heating scan with the help of the theoretical melting enthalpy of 408
the 100 % crystalline polymer: 93 J⋅g-1 for PLA45, 165 J⋅g-1 for PP46, 139 J⋅g-1 for PCL47, 138.6 409
J⋅g-1 for PVA48. Glass transition temperatures of all polymers except PVA were measured in the 410
second heating scan and taken at the mid-point of the heat capacity step. In the case of both dry 411
and plasticized PVA, Tg values were determined from the first heating scan. Determinations 412
were triplicated. 413
414
3.3 Methods 415
Diffusion coefficient determination 416
According to expected values of diffusion coefficients, roughly below and above 10-14
m2⋅s-1, 417
two complementary solid-contact methods operating at two different length scales were used to 418
reach contact times shorter than two weeks. It was checked that both methods gave similar 419
diffusion coefficients for the solute common to both series: biphenyl. Films acting as sources of 420
solutes were formulated with each solute either by soaking films in a 0.05 g⋅ml-1
solute-ethanol 421
solution during a minimal duration of one week at 60°C (cases of PP, PVA) or by adding the 422
desired solute to the casting solution at a concentration of 0.2 wt% (case of PCL). Due to the 423
difficulty of absorbing bulky aromatic solutes in PLA films below or close to its Tg (to avoid 424
recrystallization), PP films were used as sources instead in PLA experiments. All processed films 425
were stored stacked to prevent solute losses and to facilitate the internal homogenization of 426
concentration profiles. The uniformity of concentration profiles in sources was tested over the 427
cross section of microtomed films by fluorescence imaging. 428
429
-
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A modified method originally proposed by49
was used for high diffusion coefficients (above 430
10-14
m2⋅s-1, i.e. mainly in a temperature range of Tg+90 K and Tg+110 K). It consisted in 431
stacking twelve virgin films with two source films formulated with the considered solute. 432
Theoretically, by positioning sources in positions 5 and 10 in a stack consisting of 14 films with 433
approximately the same thickness, the solute concentration profile evolve with diffusion time 434
from a bimodal one to a monomodal one (with one single maximum located between films 7 and 435
8). The variations in shape of the profile combined with the variations in concentration improved 436
dramatically the comparison with the corresponding theoretical profiles expressed in function of 437
the dimensionless position x/l and dimensionless time Fo=Dt/l2, where x is the position and l the 438
average film thickness. In our experiment, films cut as disks were folded in aluminum foil and 439
inserted in a copper cylinder of a same diameter. The cylinder was closed by two Teflon disks 440
and the whole stack was packed with pressure by a screw system. Such conditions ensured good 441
contact between films without mass transfer resistance and impervious conditions at both ends of 442
the stack and on its lateral surface. After contact time at a constant temperature, solute 443
concentration in each film was measured after solvent extraction in double beam UV/VIS 444
spectrophotometer (model UVIKON 933, KONTRON Instruments, France). Extraction solvents 445
were dichloromethane for PP and PCL, and deionized water for PVA. D values were retrieved by 446
fitting numerically dimensionless theoretical concentration profiles (including the real geometry 447
of each film) to measured concentration profiles. When several contact times or stack geometries 448
were used for the same solute, D was defined as the regression slope of Fo versus x/l2. 449
450
For low diffusion coefficients, similar principles but at microscopic scale were used to reach 451
similar contact times. A source film was sandwiched between two virgin films during a 452
-
Page | 24
prescribed time. Diffusion was stopped by quenching films in liquid nitrogen, The concentration 453
profile along the section of each film was subsequently determined, after microtoming (model 454
LKB 2218 HistoRange, LKB-Produkter AB, Sweden) with cutting thickness of 15 µm, by 455
deep-UV fluorescence microspectroscopy on an inverted microscope (model IX71, Olympus, 456
Japan) in epi-configuration mode. A synchrotron source with a specific excitation wavelength 457
ranging from 275 nm to 295 nm was used according to the tested solute (DISCO beamline, 458
synchrotron Soleil, France). Bi-dimensional fluorescence emission spectra were acquired with a 459
spectral resolution of 0.5 nm between 280 nm and 480 nm (LSM 710, Zeiss, Germany), and a 460
spatial resolution of 0.5 or 1 µm. Concentrations were inferred from the surface area between the 461
spectrum and the baseline. D values were identified similarly by a fitting procedure with a 462
numerical model incorporating partitioning effects when required. Three to five concentration 463
profiles along the same section were used in the fitting procedure. 464
All experiments were triplicated or duplicated. All numerical models relied on finite volume 465
difference scheme taking into account the real thickness of each film and a solver optimized for 466
diffusion problems and distributed as an open-source project50
. 467
468
Constrained molecular dynamics simulation 469
Displacements of solute atoms available in constrained oligophenyls solutes were studied 470
regardless the polymer host by constrained molecular dynamics. Starting from a random 471
configuration, explicit constraints were applied by fixing the positions of all atoms of one or 472
several phenyl rings. Long-term molecular dynamics simulations of isolated molecules subjected 473
to such constraints were performed in the NVT ensemble at 298 K under vacuum boundary 474
conditions using a Nosé-Hoover thermostat. For covalent and non-covalent interactions, 475
-
Page | 25
COMPASS forcefield51
was employed without any cutoff and dynamic simulations were 476
conducted with Discover program (Accelrys, USA). Mean-square displacements of the center-of-477
mass and covariances of atom displacements were averaged over several initial configurations. 478
479
4 RESULTS AND DISCUSSION 480
4.1 Comparison of the scaling of D between linear aliphatic solutes and aromatic solutes 481
The assumption of the scaling exponent α following a temperature dependence as proposed in 482
Eq. (8) is tested in Fig. 2 for α values inferred from diffusion coefficients collected at different 483
temperatures on a semi-crystalline polyethylene (PE)3, 14-16 and on amorphous polyamide, based 484
upon poly-(ε-caprolactam) lightly cross-linked with diglycidyl ether of bisphenol A5. It is 485
important to notice that reported α values are associated to diffusion coefficients measured only 486
within the same study (with the same polymer and the same measurement protocol) and for 487
solutes larger than 70 g⋅mol-1. Tg values involved in Eq. (8) were either the reported ones or 488
derived from the host polymer mass. As applied in Fig. 1, solute series were categorized as 489
linear aliphatic and aromatic solutes. The “linear aliphatic” series merged linear n-alkanes, 490
n-alcohols and esterified phenols with long n-alkyl chains (n=6..18) whereas the “aromatic” 491
series combined short n-alkylbenzenes (n=0..4), substituted and hindered phenols (M= 94-545 492
g⋅mol-1). Values of α generated by this study for oligophenyls were excluded from the fitting 493
procedure and included only a posteriori as external validation of Eq. (8). 494
-
Page | 26
495 Figure 2. Scaling exponents of trace diffusion coefficients versus T-Tg. Symbols plot 496
determinations of α inferred from experimental D values reported in references enclosed within 497
brackets. Values for oligophenyls measured by this study were not used to fit Eq. (8). The 498
classification of solutes as linear aliphatic and aromatic is based on the same one as used in Fig. 499
1. 500
501
Eq. (8) fitted well α values of both linear aliphatic solutes and aromatic solutes and also offers 502
a continuous reconciliation between the diffusion behavior in solids and melts. Master curves 503
looked homothetic with a positive temperature shift of 91 K from linear aliphatic solutes to 504
aromatic solutes, corresponding to a Kβ value of 40 K and -51 K respectively. At high 505
temperature, a unitary coefficient α, as predicted by the Rouse theory13
, has been found by von 506
Meerwall et al.3 for linear aliphatic solutes, but it has not still been described for aromatic 507
solutes. The similar sharp decrease observed of aromatic solutes suggests, however, that an 508
exponent close to unity could be universal for both linear aliphatic and aromatic solutes in 509
-
Page | 27
aliphatic polymers and by extension in elastomers and thermosets far above their Tg. At 510
intermediate temperatures (in semi-crystalline or below the flow threshold of the polymer), 511
exponents increase dramatically when the temperature is decreasing, with literature values 512
reported up to 4.75. The results obtained in this study on oligophenyls enabled an external 513
validation of Eq. (8) applied to aromatic solutes near Tg where no data has been published. The 514
comparison showed a good agreement whatever the considered aliphatic polymer, PP and PCL, 515
and demonstrated the existence of α values even larger than previously reported ones. 516
517
As no existing theory holds for exponents α larger than 1 and solutes that are non-entangled 518
with polymer segments, some analogies with the scaling of self-diffusion in monodisperse 519
systems are first discussed. In monodisperse mixtures of non-entangled n-alkanes (M=114-844 520
g⋅mol-1), scaling exponents were described to decrease almost linearly from 2.72 to 1.85 when 521
temperature was increasing from 303 K to 443 K52, 53
. Tg of corresponding liquid n-alkanes are 522
expected to be lower than polyethylene (theoretically 200 K for an infinitely long 523
polyethylene54
), with values ranged between 51 K and 186 K (according to the equation of Fox 524
and Loshaek55
and parameters fitted in56, 57
). According to Fig. 2, self-diffusion could be 525
envisioned equivalently as trace diffusion in a host with a very low Tg. In a small range of 526
temperatures centered around a reference temperature θ, the linear decrease of α(T) is 527
particularly granted by the asymptotic behavior of Eq. (8) towards the melt region. When 528
max 0,T Tg K , Eq. (8) is indeed approximated by529
2
1T K T Tg K T T Tg . The absence of effect of Kβ far from Tg 530
suggests that only thermal expansion effects of the polymer host dominate. They are controlled 531
by the value of Kα, which was found very similar for both linear and aromatic solutes, 144 K and 532
-
Page | 28
156 K respectively. Similar arguments were used to explain the thermal dependence on α in non-533
entangled monodisperse systems52, 53
.They may be nevertheless approximate because not only 534
the static properties (i.e. density, free volumes distribution… of the polymer) are affected by the 535
host molecular mass but also the dynamic ones (i.e. Tg is increasing with M)3, 4
. 536
Near Tg, the variation of static polymer effects cannot be invoked alone to justify the large 537
values of α. We related phenomenologically this additional effect in Eq. (8) to a guest parameter, 538
formally -Kβ, which can be envisioned as a critical temperature deviation to Tg to translate an 539
elemental jumping unit. For solutes consisting in the repetition of a similar jumping unit, Kβ was 540
thought to be constant: positive when the jumping unit resembles polymer segments and can 541
easily accommodate the fluctuations of the contour of polymer segments; and negative 542
otherwise. The concepts of accommodation between a bulky guest molecule (e.g. aromatic 543
fluorescent dye) and host aliphatic chains has been studied in low molecular weight alkanes58
. 544
Coarse grained simulation of molecular dynamics of spherical solutes larger than the polymer 545
beads39
confirmed further the proposed description involving both polymer static and dynamic 546
effect: trace diffusion coefficients were found scaled as a power law of the volume of the bead 547
with a scaling exponent increasing from 0.8 to 1.43 when the stiffness of the polymer host 548
increased. 549
550
4.2 Scaling diffusion coefficients according to Eqs. (1), (17)-(18) 551
Phenomenological scaling of D with M at different temperatures for both tested aromatic 552
solute series is depicted in Fig. 3 along with the predictions according to Eqs. (17) and (18). One 553
important goal is to demonstrate that the temperature shift factor associated to D depends on 554
some solute contributions and that proposed equations prolong naturally the conventional 555
-
Page | 29
Williams-Landel-Ferry model (see Eqs. 9 and 15). To test the proposed free volume theory, the 556
following fitting procedure was applied. Polymer related parameters, Ka and Kb, were 557
exclusively fitted from the D values of diphenyl alkanes series at constant temperature (i.e. from 558
D values obtained in different polymers with different Tg). The determined values of Ka and Kb 559
values were directly applied to each oligophenyl solute to extract Easolute(NPh) and Ssolute(NPh). 560
External validation was finally achieved by predicting D values of biphenyl in plasticized and 561
unplasticized PVA, where the experimental D values of PVA were evidently not included in the 562
fitting procedure. 563
Fig. 3 shows the very strong mass dependence on D values for both diphenyl alkanes and 564
oligophenyls, with α decreasing with increasing temperature from 24 to 20 (95% confidence 565
interval ∓ 4.5), and from 5.3 to 4.3 (95% confidence interval ∓ 1.6), respectively. Such values 566
were far from values previously reported for aromatic solutes (see Fig. 2) and varied in a small 567
extent with tested temperatures and considered polymers. As depicted in Fig. 4, Eqs. (17) and 568
(18) fitted well the broad distribution of D over five decades for all tested polymers, with a 569
fitting error distributed normally and in the range of experimental errors. Predictions of D values 570
of biphenyl in an external polymer (PVA) by Eq. (17) were also in good agreement with 571
experimental data for both dry and plasticized PVA. The predictions for dry and plasticized PVA 572
confirmed the temperature-humidity induced plasticization superposition assessed with 573
fluorescent diffusion probes in polyamide59
. Such preliminary comparisons between 574
experimental values and model ones justify globally the separation of the polymer and solute 575
contributions for aromatic solutes in aliphatic polymers. Both effects are analyzed separately 576
hereafter. 577
-
Page | 30
578
Figure 3. a) Log-Log plot of trace diffusion coefficients of diphenyl alkanes and b) oligophenyls 579
in various polymers. Symbols are experimental values. Continuous straight lines are scaling 580
relationships fitted according to Eq. (1). Dashed lines are values fitted from Eqs. (17) and (18) 581
for diphenyl alkanes and oligophenyls respectively. 582
-
Page | 31
583
Figure 4. Comparison between calculated (with Eqs. (17) and (18) respectively) and measured 584
diffusion coefficients of a) diphenyl alkanes series and b) oligophenyls series in PLA, PP, 585
PCL(×) and PVA(▲).The continuous lines plot the straight line y=x. The corresponding 586
distribution of relative fitting errors values and fitted Gaussian distribution (continuous curve) 587
are plotted within insets. Values in PVA are external validation predictions not used in the fitting 588
procedure. 589
590
4.3 Polymer effects as probed with diphenyl alkanes 591
Diphenyl alkanes series presented several remarkable features to probe polymer effects. 592
Firstly, solutes corresponding to a small range of NC values (here 0, 1, 2) are of similar size so 593
-
Page | 32
that they are probing almost the same size of free volume pockets. Secondly, increasing the 594
distance between phenyl rings enabled to assess the effect of expected higher correlations with 595
the surroundings, C(t). Finally, the large spread of diffusion coefficients with NC improved the 596
accuracy on polymer parameter estimates, Ka and Kb, used in Eqs. (17) and (18). 597
By noting , 0 exp apolymer Cb
KD D T N
K T Tg
, Fig. 5 plots both the scaling of D and 598
the solute contribution defined as D/Dpolymer versus the number of carbon atoms, NC, at a constant 599
temperature. The inferred scaling did not depend on the considered polymer and was associated 600
to a 10C
N value of 1.3 in Eq. (17), which implies that D decreases 10-folds when 1.3 carbons is 601
added between the two phenyl rings. 602
603
-
Page | 33
Figure 5. Scaling diffusion coefficients of diphenyl alkanes with the number of carbons, NC, 604
between phenyl rings: a) raw diffusion coefficients measured at 343 K for PLA, 333 K for PP, 605
and 323 K for PCL b) diffusion coefficients relative to biphenyl , 0CD T N . Eq. (17) 606
predictions are plotted as continuous lines. 607
608
The corresponding polymer contribution was assessed as D/Dsolute with 609
10
, 0 exp ln10 Csolute CC
ND D T N
N
. D and D/Dsolute in polymers used for fitting (PLA, PP, 610
PCL) and validation (PVA) are plotted versus T-Tg in Figs. 6a and 6b respectively. Results 611
showed accordingly an evolution of D/Dsolute that was independent of the considered solute and 612
where polymer effects were finally reduced to an effect of the distance to Tg. Values of Ka and 613
Kb were found equal to 600 K and 58 K respectively, and predicted independently with an 614
acceptable accuracy of the D values of biphenyl in both dry and plasticized PVA. The estimated 615
value of Kb is of the same magnitude order as the value of 50 K for the polymer-related 616
parameter K22 reported in30
, when polystyrene is probed with toluene and ethylbenzene. As 617
reported in32
, such ranges of Kb were assumed to be generic for linear polymers and therefore 618
also valid for oligophenyls series too. According to Eqs. (9) and (15), it could be thought that 619
(Ka, Kb) (i.e. fitted on our diphenyl alkanes data) and (Kα, K ) (i.e. fitted exclusively from 620
literature data on different aromatic solutes) might be also related together. However, it is 621
expected to be true only if no additional energy barrier exists. In our study, it is very likely for a 622
solute comprising only one single jumping unit. By taking the molecular mass of benzene 623
(M/M0=78 with M0=1 g⋅mol-1
) and the average value of Kα reported in section 4.1 (150 K), we 624
-
Page | 34
get a value Kα⋅lnM of 654 K close to the value of Ka (ca. 600 K) reported here. The agreement 625
between Kb and K is even more convincing with values of 58 K and 51 K, respectively. 626
627
Figure 6. Experimental diffusion coefficients of diphenyl alkanes in PLA, PP, PCL (empty 628
symbols) and PVA (filled symbols) versus T-Tg a) raw diffusion coefficients, b) D/Dsolute. Eq. 629
(17) fitted on empty symbols is plotted as continuous lines. Filled symbols are used for external 630
validation purposes. 631
632
4.4 Solute activation parameters of oligophenyls 633
Diffusion coefficients of oilgophenyls exhibited much a lower dependence with molecular 634
mass, which was associated in Eq. (18) to a free energy barrier, which is also a function of the 635
-
Page | 35
number of phenyl rings. Fig. 7 plots the dependence of D (Figs. 7a and 7b) and of D scaled by 636
the polymer exponential factor, exp a
b
K
K T Tg
, (Figs. 7c and 7d) versus the number of 637
phenyl rings, NPh. Whatever the considered temperature and tested polymer, the exponential 638
decrease of D with NPh was non-regular, suggesting a non-monotonous variation of the energy 639
barrier to translation with NPh. The trend is confirmed by plotting both D and D scaled by 640
polymer effects on a van’t Hoff plot in Fig. 8. While a significant deviation to an Arrhenius 641
behavior was observed on raw diffusion coefficients in polymers close to their Tg as shown in 642
Fig. 8a and 8b, a pure Arrhenius behavior was recovered by contrast once the effects of density 643
were corrected. It was particularly interesting to notice that the slope of the van’t Hoff plot was 644
systematically much higher for p-terphenyl than those of the former and following solutes in the 645
series. 646
-
Page | 36
647
Figure 7. Scaling of diffusion coefficients of oligophenyls with the number of phenyl rings, NPh, 648
a,c) in PCL and b,d) in PP at different temperatures: a,b) raw diffusion coefficients, c,d) scaled 649
diffusion coefficients with polymer effects removed. Predictions according to Eq. (18) are 650
plotted as continuous lines. 651
-
Page | 37
652
Figure 8. Normalized van’t Hoff plots of oligophenyls a,b) raw diffusion coefficients and c,d) 653
scaled diffusion coefficients with polymer effect removed a,c) in PCL and b,d) in PP. 654
Predictions according to Eq. (18) are plotted as continuous lines. 655
656
The non-monotonous variations of solute activation energies, Easolute, and entropies, Ssolute, are 657
specifically captured in Figs. 9b and 9d. Raw activation values, as estimated from Figs. 8a and 658
8b, are also given in Figs. 9a and 9c. Regardless polymer effects were included or not, activation 659
parameters exhibited systematically a hat shape with a maximum for NPh=3. Such 660
non-monotonous variations appeared for both tested polymers (PP and PCL) with apparent 661
activation energies and entropies of PP shifted from PCL ones by approximately 35 kJ⋅mol-1 and 662
-
Page | 38
95 J⋅mol-1⋅K-1, respectively. PP and PCL activation values were exactly reconciled once 663
polymer effects were removed by Eq. (18). It is worth to notice that the reconciliation was 664
obtained by removing the polymer contribution, whereas it was inferred independently from the 665
D values of diphenyl alkanes. Since both enthalpy and entropy exhibited similar shape, an 666
apparent position correlation between both quantities was found as plotted in Fig. 9e. Enthalpy-667
entropy compensation is often considered to be a statistical artifact due to correlations between 668
errors on each estimate and because entropy must be extrapolated to infinite temperature. As 669
polymer effects were removed in our case, it could however be thought that the extrapolation of 670
solute effects regardless the true physical state of the polymer at an infinite temperature can have 671
a reasonable meaning. Such kind of discussions can be found in60
. The corresponding internal 672
free energy barriers at 298 K required to induce a translation of oligophenyls are represented in 673
Fig. 9f. As activation energy and entropy contribute with opposite signs to the free energy 674
barrier, free energy was found monotonous with NPh. 675
-
Page | 39
676
Figure 9. a,c) Raw and b,d) solute activation energies and diffusion entropy of oligophenyls in 677
PP and PCL versus the number of phenyl rings, NPh; e) correlation between solute activation 678
parameters; f) related free barrier energy to diffusion at 298 K. 679
680
4.5 Mechanisms of translation of oligophenyls in aliphatic polymers 681
Diffusion of large organic solutes in polymer hosts cannot be directly studied by molecular 682
dynamics close to their Tg at atomistic scale. Our experimental results highlighted however two 683
-
Page | 40
noticeable properties to derive tractable simulations and to gain further insights on the translation 684
mechanisms of linear aromatic molecules in rubber polymers: 685
- The displacements of each jumping unit (i.e. “polymer effects” in the text) and “solute 686
effects” (i.e. collective displacements of jumping units) on D are separable; 687
- Activation energies associated to the displacement of several jumping units vary with the 688
parity of the number of jumping units. Such a feature can be directly tested by simulation. 689
The main idea was to test whether two correlation modes between the displacements of phenyl 690
rings and surrounding atoms could explain the effect of the parity of NPh on activation terms. 691
Since neither the partitions between both modes nor the correlation times are known, our strategy 692
consisted in studying the translation of CM in constrained long-term molecular dynamics (10 ns 693
or more), where the positions of one or several phenyl rings (i.e. jumping units) are kept fixed, 694
under vacuum boundary conditions (i.e. without polymer host). The minimum times to enable a 695
translation of CM longer than a phenyl ring diameter (i.e. length of one jumping unit) and the 696
related translational entropy were analyzed according to Eqs. (19) and (24) respectively and 697
finally compared to their experimentally inferred counter-parts: Easolute and Ssolute. By fixing a 698
priori different values to p, it was in particular possible to assess which value could reproduce a 699
non-monotonous variation with the number of jumping units. 700
The results obtained by averaging over a wide range of initial configurations are plotted in Fig. 701
10 and compared to experimental values reported in Fig. 9. The typical “hat shape” of activation 702
energies was particularly reproduced without significant bias with RTln(τring-τtrapped) calculated 703
from Eqs. (19) and (20) when p→1 (Fig. 10a). Ssolute values derived from Eqs. (21)-(24) led also 704
approximately to the similar trend when p→1 (Fig. 10b). The theoretical translational entropy 705
underestimated however systematically the real one due to the loss of fluctuations information 706
-
Page | 41
caused by the fixed positions of some phenyl rings. It is worth to notice that the reported “hat-707
shape” of activation terms could not be predicted with the simple coarse-grained theory 708
supported either in Eq. (5) or in Eq. (7) because they assume that the displacements of all phenyl 709
rings are equivalent. In reality, the gyration radius and mean-square-displacement of 710
oligophenyls are dramatically affected by the combination of phenyl rings that are fixed. Among 711
all tested oligophenyls (NPh=2,3,4), p-terphenyl is the one that offers the highest ratio of 712
possibilities to reduce dramatically the fluctuations of CM and other atoms. Particularly efficient 713
configurations (i.e. five over the 23 possibilities): blocking the central ring alone or blocking 714
randomly two or three rings. Such effects could not be captured by a generic flexible model that 715
assume a uniform relaxation model along the chain (see Eq. (3)) or uniform covariances between 716
connected jumping units (see Eq. (4)) and were consequently better reproduced by atomistic 717
simulations. It is particularly noticeable that increasing p in simulations induced enthalpy-718
entropy compensation as experimentally assessed so that the partitioning between short and long-719
lived contacts could be proposed as the main cause of the phenomenon. Corollary, the suggested 720
high value of p (i.e. at least one jumping unit has long-lived contacts with surrounding host) does 721
not depend on the length of studied solutes and could be general for all linear aromatic solutes 722
within aliphatic polymers. 723
-
Page | 42
724
Figure 10. a) Comparison of relative solute activation energy (continuous lines, left scale), 725
calculated as ring rinPh Phg trappedN N for different p values according to Eqs. (19)-(20), 726
with experimental values (dotted lines, right scale) reported in Fig. 9b. The horizontal dashed 727
line represents the average value of Easolute for studied oligophenyls. b) Comparison of solute 728
entropy (continuous lines, left scale), calculated for different p values according to Eqs. (21)-729
(24), with experimental values (dotted line, right scale) reported in Fig. 9d. 730
731
-
Page | 43
5 CONCLUSIONS 732
The presented work introduces several invariant scaling relationships for the trace diffusion 733
coefficients (D) of bulky solutes in thermoplastic polymers above their Tg and presents a first 734
molecular interpretation of corresponding translation mechanisms. D values over five decades 735
were measured for two homologous series of short aromatic solutes, diphenyl alkanes and 736
oligophenyls, in different polymers and at various temperatures, T. The collected values 737
completed the available picture from literature by integrating D values for low T-Tg values. The 738
analysis of the scaling with molecular mass, as D∝M-α(T-Tg), shows that aromatic solutes have a 739
parallel behavior to linear aliphatic solutes but shifted by ca. +91 K. The proposed scaling is 740
notably able to reconcile α values experimentally assessed in melts3 and in solids. It could be 741
thought that the temperature shift would include the behavior all organic solutes larger than 742
voids, with left and right bounds set by linear aliphatic and aromatic solutes respectively. 743
However, α values derived for diphenyl alkanes showed that aromatic solutes including a 744
flexible segment between two phenyl rings were associated to a much larger temperature shift. 745
Such dramatic effect of the solute chemical structure was found independent of the considered 746
aliphatic polymer and associated only to the size and type of the solute jumping unit. It would 747
explain both the logarithm dependence of the apparent activation energy with molecular mass 748
and the additional deviation to the Arrhenius behavior close to Tg, as specifically discussed in4. 749
Because the specific volume of polymers is also a function of T-Tg, a modification of the 750
distribution of free volume in the polymer must be also invoked, as already suggested for the 751
self-diffusion of non-entangled n-alkanes52
. 752
To reach a consistent description of all effects, an extended free volume theory is proposed 753
assuming that the free volume fluctuations of polymer host segments, which control the short-754
-
Page | 44
range translation of each jumping unit and finally the displacements of many jumping units obey 755
to two independent statistical distributions with different activations by temperature. Static 756
polymer effects were associated to a single exponential term, exp a bK T Tg K . It was 757
verified thus that the values of 600 K and 58 K, proposed for of Ka and Kb respectively, enabled 758
the extrapolation of known diffusion coefficients from one aliphatic polymer to another one (e.g. 759
unplasticized to plasticized, apolar to polar), at least at a similar temperature. If the diffusion 760
coefficient needs to be extrapolated to a different temperature, the solute-related activation needs 761
to be accounted specifically. Solute effects were found to follow an Arrhenius’ law for 762
oligophenyls including from two to four phenyl rings. Related activation energies and entropies 763
were, however, highlighted to vary non-monotonously with M. 764
This outstanding effect of the parity of the number of ring was used to assess, via constrained 765
molecular dynamics simulations, the probability of phenyl rings (jumping units) to behave as 766
anchors in the translation mechanisms of aromatic solutes. It is argued that, though the polymer 767
at rubbery state does not control the reorientation frequency of the entire molecule (i.e. each 768
phenyl ring is displacing independently while being limited by the connectivity of rings), it tends 769
to hinder the reorientation of individual phenyl rings. Our simulations show that constraining 770
randomly one or several phenyl rings slows down the mean-square-displacement of the center-771
of-mass (CM) of solutes in a different way according to the number of rings, NPh, being odd or 772
even. For example, when NPh=3, it is twice more likely to block a ring at one end than in the 773
middle; so that the slowdown is stronger than for NPh=2 or 4. As our interpretation matched 774
remarkably the relative energies experimentally determined on oligophenyls series, it is thought 775
that the proposed anchor effect of phenyl rings is universal in aliphatic polymers at least between 776
-
Page | 45
Tg+51 K and Tg+150 K, where α was estimated to be much larger than unity (as shown in Fig. 777
2). 778
Presented results should find applications in many domains where diffusion coefficients of 779
aromatic molecules and polymer residues are particularly critical: contamination by substances 780
leeched from polymer materials, loss of additives during physical ageing of polymers, 781
reactivities in polymers in processing and use conditions. In particular, reported results open the 782
way to design of substances with low diffusion coefficients with an odd number of rings and/or 783
with flexible segments close to CM. The behavior of branched aromatic molecules will be 784
presented in a companion paper. 785
786
6 ACKNOWLEDGEMENTS 787
The first authors would like to acknowledge the support of the PhD grant from Région Île-de-788
France. We also gratefully thank Dr. Frédéric Jamme for his technical assistance. This work was 789
supported by the DISCO beamline of the synchrotron Soleil (Proposal 20100909). 790
791
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For Table of Contents use only
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Diffusion of aromatic solutes in aliphatic polymers above glass 884
transition temperature 885
Xiaoyi Fang2, Sandra Domenek
2, Violette Ducruet
1, Matthieu Refregiers
3, Olivier Vitrac
1* 886
887
888
889