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H* and monomer volume and correlation lengths for glass-formers
Journal of Non-Crystalline Solids 357 (2011) 404–410
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Journal of Non-Crystalline Solids j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / j n o n c r y s o l
The defect diffusion model, scaling, EV*/H* and monomer volume and correlation lengths for glass-formers J.T. Bendler a, J.J. Fontanella b,⁎, M.F. Shlesinger b,c, M.C. Wintersgill b a b c
BSC, Inc., 3046 Player Drive, Rapid City, SD 57702, USA Physics Department, U.S. Naval Academy, Annapolis, MD 21402, USA Office of Naval Research, ONR Code 30, 875 N. Randolph St., Arlington VA 22203, USA
a r t i c l e
i n f o
Article history: Received 19 January 2010 Received in revised form 29 June 2010 Available online 21 August 2010 Keywords: Dielectric relaxation; High pressure; Scaling; Correlation length
a b s t r a c t Four topics are treated within the framework of the defect diffusion model (DDM). First, it is shown how the relationship between EV*/H* (ratio of the apparent isochoric activation energy to the isobaric activation enthalpy) and monomer volume for polymers that has been pointed out by Floudas and co-workers [G. Floudas, K. Mpoukouvalas and P. Papadopoulos, J. Chem. Phys. 124 (2006) 074905] is predicted. Next, it is shown that in the DDM, scaling arises because the critical temperature can be represented approximately by a power law. Consequently, in the DDM scaling is always approximate and significant departures from scaling, as is observed in the case of hydrogen bonded materials for example, are matters of degree. It is also shown how the connection of scaling with EV*/H* is a natural consequence of the DDM. Finally, DDM calculations of the defect correlation length are carried out and compared with experimental dynamical correlation lengths measured using the 4D3CP solid state NMR method. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The effects of volume and temperature on the properties of glassforming liquids are topics of current interest [1–4]. In order to separate these effects, the ratio of the apparent isochoric activation energy, EV or EV*, EV = EV* =
∂ lnx ∂ð1=T Þ V
ð1Þ
to the isobaric activation enthalpy, EP or H* EP = H* =
∂ lnx ; ∂ð1=T Þ P
ð2Þ
where x is a transport property such as electrical relaxation time, viscosity or ionic conductivity, is often considered. This ratio can be written EV* = H*
∂ ln x ∂T V
=
∂ ln x : ∂T P
ð3Þ
Many experimental evaluations of this ratio have been reported [3,5] and the authors have recently derived an expression for the ratio ⁎ Corresponding author. Tel.: + 1 410 293 5507; fax: + 1 410 293 5508. E-mail addresses: [email protected] (J.T. Bendler), [email protected], [email protected] (J.J. Fontanella), [email protected] (M.F. Shlesinger), [email protected], [email protected] (M.C. Wintersgill). 0022-3093/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2010.07.031
based on the defect diffusion model (DDM) [6,7]. In a recent paper, Floudas et al. have established a connection between EV*/H* and the monomer volume, Vm, for polymers [3]. They find that as Vm increases, in general EV*/H* decreases. This trend is accompanied by significant deviations. In the present paper, it is shown that this connection is accounted for by the DDM. A second experimental observation related to EV*/H* is known as scaling. Specifically, it has been found that for many glass-forming liquids ln(x) scales at least approximately with 1/(TVγ) and that scaling is related to EV*/H*. A review of the work until 2005 is given in Ref. [5]. In the present paper, it is shown how the phenomena related to scaling are natural consequences of the DDM. Finally, the DDM is used to calculate correlation lengths and the results are compared with experimental dynamical correlation lengths measured using the 4D3CP solid state NMR method [8–12]. 2. Theory The basic assumptions for the DDM are that a super-cooled liquid contains mobile single defects (MSDs), immobilized clustered single defects (ICSDs) and normal liquid molecules. In the case of fragile liquids, one may consider defects as persistent packets of free volume, MSDs being mobile defects with greater than average free volume and ICSDs having less than average free volume. Defects are persistent in that they are conserved, neither created nor destroyed, and simply converted thermodynamically between MSDs and ICSDs with changes in pressure and temperature. Transitory molecular configurations with instantaneous free volumes higher or lower than average are not permanent “defects” in this sense, but rather contribute to the high
J.T. Bendler et al. / Journal of Non-Crystalline Solids 357 (2011) 404–410
frequency background fluctuations of the molecular properties of the fluid. MSDs and ICSDs are associated with regions where the density is smaller and larger than average, respectively. Consequently, MSDs are large and ICSDs are small [13]. Because ICSD-rich regions of the fluid are rigid regions and MSD-rich regions are mobile, dynamic heterogeneity is an essential aspect of the DDM picture of super-cooled liquids. Finally, all of the physical properties of a fragile super-cooled liquid are influenced by the tendency of the MSDs to cluster (i.e., to form ICSDs/rigid regions) as temperature decreases or pressure increases. In the DDM the concentration of MSDs, c1, is given by [7] 0:5η
c1 = c exp −
βKWW B TC
!
ðT−TC Þ0:5η ð1−δÞ
:
ð4Þ
This leads to the following equation for the dielectric relaxation time −1 = β
τDDM = c
! 0:5η B TC Δo exp : kB T ðT−TC Þ0:5η ð1−δÞ
τo exp
ð5Þ
Δ0 is the smallest barrier opposing a defect hop. The exponential containing Δ0 together with the prefactor τo , give the shortest pausing time for the jump of a defect [14,15]. As usual, the Arrhenius term will be omitted so that the working equation is τDDM ≈c
−1 = β
B TC
0:5η
τo exp
!
ðT−TC Þ0:5η ð1−δÞ
:
ð6Þ
In Eqs. (5) and (6), c is the total concentration of defects (i.e., the fraction of lattice sites occupied by a defect), τo is the shortest time for the jump of a defect, and βKWW is the Kohlrausch–Williams–Watts or stretched exponential parameter associated with the relaxation function [16,17] β ϕðt Þ = A exp −ðt =τDDM Þ KWW :
ð7Þ
(1 − δ) = V(T,P)/Vo and V(T,P) is the volume of the liquid at pressure P, and absolute temperature T, and Vo is a reference volume at P = 0 at a given reference temperature. For consistency with the notation given in other papers in the literature, for the remainder of this paper, V(T,P)/ Vo will be referred to simply as V. η is an integer (1, 2 or 3) describing the dimensionality of the correlation volume [18]. If δ = 0 and η = 3, a 3/2 power law results. Materials such as poly(propylene glycol) are successfully described by η = 3 [13]. If, on the other hand, the correlation volume grows in a two-dimensional manner, with η = 2, then Eq. (6) reduces to the standard Vogel equation [19], (assuming that volume is not strongly dependent upon pressure or temperature i.e. δ = 0). For example, glycerol is best described by η = 2 [13]. A correlation length, ξj,DDM, corresponding roughly to the distance beyond which the defects do not interact, is given by [14,18] ξj;DDM ðT Þ = Lj
TC T−TC
0:5
:
ð8Þ
TC is the critical temperature at which all defects would be clustered ICSDs if the glass transition did not intervene. The critical temperature, TC, depends on pressure, i.e. TC = TC(P). (See also Eq. (13) below where a simple Bragg–Williams lattice model for TC is discussed.) The correlation length of Eq. (8) plays a central role in the DDM analysis, and, in particular, in the derivation of the VTF-like temperature laws such as Eq. (6) [20,21]. As will be shown, this quantity appears to be associated with experimental dynamical correlation lengths. As the temperature falls, more defects become immobile, locked into clusters and thus unavailable to promote relaxation. Only mobile single (uncorrelated) defects (MSDs) remain effective in producing relaxa-
405
tion, and this depletion of MSDs with falling temperature leads directly to Eq. (6). Another parameter in the exponent of Eq. (6) is
B =−
L1 L2 L3 lnð1−cÞ : d3o βKWW
ð9Þ
do is the average distance between neighboring sites at P = 0 and at a given reference temperature. The three indices on the “direct” correlation lengths Lj allow for anisotropic defect–defect interactions. If the interactions are isotropic, then L1 = L2 = L3 = L. As has been pointed out [7], Eq. (6) is only applicable between TB (or TLL) and Tg. It is not valid below Tg because rigidity percolates below Tg and the motion of the MSDs is thereby restricted. In addition, Eq. (6) is not expected to apply to temperatures above TB (or TLL) which is on the order of 1.2 Tg. For temperatures above TB, we assume that jamming caused by the ICSD-rich regions is greatly diminished and sufficient free volume becomes available to allow new relaxation processes in addition to those controlled by MSD hopping diffusion. Consequently, the properties of the liquid will vary differently with temperature and pressure above and below TB but the changes in behavior are likely to be continuous and small. In fact, property changes near TB are often subtle, being observable primarily via the temperature dependence of a property rather than as a discontinuous change in a property itself [22]. As shown elsewhere [7], if the isothermal compressibility, κT, and isobaric volume thermal expansion coefficient, αP, are defined in the usual manner, κT = −
1 ∂V V ∂P T
ð10Þ
and 1 ∂V ; V ∂T P
αP =
ð11Þ
the assumption that TC = TC(P) leads to
EV* H*
αP T ∂TC κT TC ∂P T = : α ðT−TC Þ 1+ P 0:5η 1−
DDM
ð12Þ
(An approximate version of Eq. (12) was given in an earlier note [6].) Eq. (12) has been used to predict the ratio EV*/H* and its temperature dependence for several materials [7] and good agreement between theory and experiment is found. An interesting form of Eq. (12) is obtained using a simple Bragg– Williams treatment of a (defect) phase separation transition (for nearestneighbor pair interactions with equal occupancy of A and B sites). It has been pointed out that the critical demixing temperature is given by [23] TC =
zjΔhj zðjΔε + PΔvjÞ = 4kB 4kB
ð13Þ
where z is the lattice coordination number, kB is Boltzmann's constant, and Δh = Δε + PΔv is the decrease in enthalpy resulting from the formation of a defect pair: defect
+
defect ↔
ðdefectÞ2
ð14Þ
where Δε is the decrease in pair energy and Δv is the decrease in volume. From Eq. (13) one finds ð∂TC =∂P ÞT ≈zjΔvj = 4kB
ð15Þ
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J.T. Bendler et al. / Journal of Non-Crystalline Solids 357 (2011) 404–410
so that the controlling molecular parameter is the net (negative) volume change, Δv, caused by defect pairing. If Δv is negative (as we assume for a fragile liquid), then a pressure increase (or volume decrease) causes mobile single defects (MSDs) to cluster (and rigidity to increase) as the equilibrium in Eq. (14) shifts to the right. Substituting Eqs. (13) and (15) into Eq. (12) gives
α T Δv 1− κP T Δh
EV* H*
= DDM
1+
αP ðT−TC Þ 0:5η
:
ð16Þ
This shows that the main factor determining the value of EV*/H* is the ratio of the volume change to the change in enthalpy when defects cluster. 3. Correlation of E*V/H* with monomer volume In this section it will be shown that Eq. (12) implies that for polymers, as the monomer volume, Vm, increases, EV*/H* decreases. For simplicity, it will be assumed that η = 2 and this (Vogel) approximation give rise to the following equation.
EV* H*
α T Δv 1− κP T Δh T C ∂P = = 1 + αP ðT−TC Þ 1 + αP ðT−TC Þ α ∂T 1− κ P TT C
DDM
ð17Þ
We first focus on Δν. If it is assumed that the rigid regions are “crystal-like” and the mobile regions are “liquid-like,” Δν can be estimated from the Clapyron equation:
dP dT
= coexist
ΔSmelt ΔVmelt
ð18Þ
First, the total volume of the material is written as VT = Nm Vm + Vf + Vdefect :
ð23Þ
Nm is the number of monomers, Vm is the volume of a monomer, Vf is the free volume not associated with defects and Vdefect is the total volume of the defects. Defining V IC to be the average volume of an ICSD and V MS to be the average volume of an MSD, it follows that Vdefect = NIC V IC + NMS V MS
ð24Þ
where NIC is the number of ICSDs, and NMS is the number of MSDs. Next, if the simplest case is assumed where two MSDs form a single ICSD Δv = 2 V MS −V IC :
ð25Þ
Written in this manner, Δv is a positive value. The thermal expansion coefficient can be calculated from αP =
1 ∂ Nm Vm + Vf + NIC V IC + NMS V MS VT ∂T
ð26Þ
Next, it is assumed that the most temperature dependent terms in Eq. (26) are NIC and NMS. This is a reasonable assumption because Nm is not temperature dependent. The remaining volumes, Vm, Vf, V IC and V MS are somewhat temperature dependent, but the variation with temperature is expected to be small compared with that for the numbers of defects. Consequently, αP =
1 VT
∂N ∂NIC + V MS MS V IC ∂T ∂T
ð27Þ
Again assuming that two MSDs form one ICSD, where the quantity on the left is the slope of the solid–liquid coexistence line, ΔSmelt is the entropy change on melting and ΔVmelt is the volume change on melting. If the heat absorbed on melting (or release on freezing) is measured, then ΔSmelt can be determined from ΔSmelt =
ΔHmelt Tmelt
ð19Þ
−2
Δv≈ΔVmelt
αP =
∂NIC 1 Δv ∂NIC =− V IC −2V MS VT VT ∂T ∂T
:
ð29Þ
Alternatively, αP can be written ð20Þ
As a consequence, Δν can be estimated in several ways: (a) ΔVmelt can be determined directly from the density of the liquid and the density of the crystalline solid along with use of the approximation Δν≈ ΔVmelt; (b) ΔVmelt can be determined directly from Eqs. (18) and (19)) along with use of the approximation Δν ≈ ΔVmelt; (c) If the slope of the crystal–liquid line, (dTm/dP) is not known, then the approximation 1 dTg 1 dTmelt ≈ Tg dP Tmelt dP
ð28Þ
Consequently, the thermal expansion coefficient is
Finally, if it is assumed that Δν ≈ ΔVmelt, ΔHmelt dTmelt = : Tmelt dP coexist
∂NIC ∂NMS = : ∂T ∂T
αP = +
1 Δv ∂NMS 2 VT ∂T
ð30Þ
The numbers of defects can be written in terms of the defect concentrations as [13] NIC =
N ðc−c1 Þ 2
ð31Þ
or ð21Þ
NMS = Nc1
ð32Þ
can be used along with Eq. (20). The validity of the approximation represented by Eq. (21) has been discussed elsewhere [15,24]; (d) In addition, the following approximation
where N is the number of lattice sites. Consequently, the thermal expansion coefficient can be rewritten as
4k dTC 4k T dTg ≈ B C Δv≈ B z dP z Tg dP
αP = +
ð22Þ
could be used where z has a value from 8 to 12; (e) For the present work, the following approach based on the thermal expansion coefficient is taken.
N Δv ∂c1 2 VT ∂T
ð33Þ
(This follows from either Eq. (29) or (30).) Next, using the assumptions that lead to the Vogel equation, Eq. (4) becomes c1 = c expð−βKWW B*TC = ðT−TC ÞÞ
ð34Þ
J.T. Bendler et al. / Journal of Non-Crystalline Solids 357 (2011) 404–410
407
so that ∂c1 β B*T = c KWW 2C expð−βKWW B*TC = ðT−TC ÞÞ ∂T ðT−TC Þ
ð35Þ
This gives us the following equation for Δν Δv =
2αP VT Nc
βKWW B*TC ðT−TC Þ2
expð−βKWW B*TC = ðT−TC ÞÞ
ð36Þ
Next, because most of the material is made up of molecules, it follows from Eq. (23) that V T e Nm V m :
ð37Þ
Also, the usual definition of the lattice (one lattice site per molecule) is used so that N ~ Nm Δv =
2αP Vm c
βKWW B*TC ðT−TC Þ2
expð−βKWW B*TC = ðT−TC ÞÞ:
ð38Þ
Finally, then, Eq. (17) can be rewritten as
EV* H*
= DDM
2 ðT−TC Þ2 Vm T P 1−2α κT jΔhjTC cβKWW B* expð−βKWW B*TC = ðT−TC ÞÞ
1−αP ðT−TC Þ
ð39Þ
This shows that, if all other parameters were the same for different materials, as the monomer volume increases, the ratio decreases. That is consistent with the empirical observation of Floudas et al. [3] that there is a general tendency for the ratio to decrease as monomer volume increases. However, it is also clear that factors other than Vm also affect EV*/H* in that a plot of EV*/H* vs. Vm does not form a single curve [3]. Eq. (39) also predicts this result because different materials exhibit different values of the characteristic parameters, c, B, βKWW, etc. 4. Scaling
Fig. 2. Experimental results of Heinrich and Stoll [27] (points) and DDM predictions (six theoretical curves based on Eq. (6)) for log(τ) vs. specific volume at various temperatures for PVAc. The symbols are: Solid triangles—393 K; Open triangles—383 K; Solid circles— 373 K; Open circles—363 K; Solid squares—353 K; Open squares—343 K.
Stoll [27], shown in Figs. 1 and 2 for PVAc at 393 K and below, were analyzed. The reasons for limiting the temperature range of the data are discussed elsewhere [7]. The authors have recently considered that data using the 3/2 power (η = 3) form of Eq. (6) [7]. For comparison and more relevance to the literature, the data were reconsidered using the DDM Vogel-form (η = 2) of Eq. (6). The values of V were determined using the Tait equation in the form dðT Þ + P V = eðT Þ 1−g ðT Þ ln dðT Þ + Po
ð40Þ
The values of the fitting parameters, d(T), e(T), and g(T) are given in Ref. [7]. In order to represent TC(P), the following empirical relation was used bP 1 = b TC ðP Þ = TC ð0Þ 1 + a
ð41Þ
A concept related to the ratio EV*/H* is scaling [5,25,26]. It has been suggested that when the relaxation time is plotted vs. TVγ, the constant (scaling parameter) γ can be adjusted so that the data form a single curve, hence the term scaling. In order to evaluate the concept of scaling within the framework of the DDM, the data of Heinrich and
According to Mpoukouvalas et al. [28], an equation analogous to Eq. (41) was first introduced by Simon and Glatzel [29]. The quantities a, b and TC(0) are fitting constants. The best-fit parameters were found
Fig. 1. Experimental results of Heinrich and Stoll [27] (points) and DDM predictions (six theoretical curves based on Eq. (6)) for log(τ) vs. pressure at various temperatures for PVAc. The symbols are: Solid triangles—393 K; Open triangles—383 K; Solid circles— 373 K; Open circles—363 K; Solid squares—353 K; Open squares—343 K.
Fig. 3. Experimental results of Heinrich and Stoll [27] (points) and DDM predictions (six theoretical curves based on Eq. (6)) for log(τ) vs. 1000/(TV2.5) at various temperatures for PVAc. The symbols are: Solid triangles—393 K; Open triangles—383 K; Solid circles— 373 K; Open circles—363 K; Solid squares—353 K; Open squares—343 K.
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J.T. Bendler et al. / Journal of Non-Crystalline Solids 357 (2011) 404–410
to be log10(Aτ) = −11.17, B* = 6.06, TC(0) = 261.5 K, and a = 1.33 GPa and b = 4.47. Figs. 1 and 2 are plots of log(τ) vs. P and log(τ) vs. V/Vo, respectively. In both figures, both the data and best-fit DDM theory are shown. Clearly, it is possible to represent all of the pressure and temperature data using the DDM. Fig. 2 makes it clear that the DDM is capable of explaining the effects of temperature at a constant volume even though it is a free volume theory. Fig. 3 is a scaling plot of the same data and theory where γ = 2.55. The data for PVAc appear to scale and the value of the scaling parameter is in agreement with Roland et al. who quote a value for PVAc of γ = 2.6 [5]. The important feature is that log(τ) for PVAc calculated using the DDM appears to scale. However, a close inspection of Fig. 3 shows that the scaling predicted by the DDM is not perfect. This is more apparent in Fig. 4 where the predictions of the DDM only (for six isothermal curves) are plotted. It is clear that there is a slight scatter in the prediction and the scatter is on the order of the width of the lines. It was not possible to achieve better scaling by adjusting γ. This shows that for PVAc, at least, scaling is an approximate concept based on the DDM. Nonetheless, the predictions of the DDM make it possible to gain some insight into the reason for scaling. Toward this end, Eq. (6) is rewritten as follows. 0
1
B 0:434B* C log10 ðτDDM Þ≈ log10 Aτ + @ A T T ðP Þ−1 V
ð42Þ
C
where Vogel behavior (η = 2) has once again been assumed. Since at least approximate scaling exists in the DDM, the behavior must be caused by the denominator of Eq. (42). In fact, in the DDM, scaling arises because in the experimental range of volumes and temperatures TC ðP Þ≈kV
−φ
ð43Þ
where k is a constant. To show that this is true, TC(P) for PVAc (Eq. (10)) was plotted vs. V using the PVT data given by Eq. (40). The results are shown in Fig. 5 for 393 K. The best-fit values of k and φ are 284 K and 1.52, respectively. Further, the parameters k and φ are only weakly temperature dependent, varying from about 284 K to 272 K and 1.52 to 1.57, respectively, over the temperature range 393 K to 343 K. Consequently, Eq. (42) can be rewritten log10 ðτDDM Þ≈ log10 Aτ +
0:434B*k TV φ + 1 −kV
ð44Þ
Fig. 5. Tα vs. V for PVAc (393 K), glycerol (280 K), meta-fluoroaniline (280 K), and propylene carbonate (280 K) along with a plot of TC vs. V for PVAc at 393 K. Also included are the best-fit power law curves (Eq. (43)).
This shows that the DDM equation for the relaxation time has the functional dependence necessary for scaling, namely τDDM = F (TVpower) where F is a function. Interestingly, for PVAc, the fit gives φ ≈ 1.5 and γ ≈ 2.55. Consequently, based on the numbers for PVAc, it is tempting to conclude that the exponent in Eq. (44) is γ, i.e. it is tempting to conclude that γ = φ + 1. However, the DDM does not suggest that. Eq. (44) shows an extra factor of kV in the denominator and V, itself, is temperature dependent. Consequently, it would not be expected that γ = φ + 1 in general. To check this result, other materials were evaluated. Two other materials that exhibit scaling, meta-fluoroaniline and propylene carbonate, were studied by Reiser et al [30]. However, Reiser et al. did not report numbers for relaxation times so TC(P) could not be determined. However, Tg(P) was reported, albeit in a slightly different form
Tg ðP Þ ≡ Tg ðP = 0Þ +
∂P
P+
2 1 ∂ Tg 2 P : 2 ∂P 2
ð45Þ
Consequently, power law behavior was studied for Tg(P) (or, more precisely, Tα(P)) rather than TC(P). In order to verify that this makes little difference, Tg(P) for PVAc from Roland and Casalini [31] was plotted vs. V and the results are also shown in Fig. 5. The value of φ was 1.64 in reasonable agreement with the value of 1.52 determined from TC(P). A value of k = 340 K was also obtained showing that Tg(P) is larger than TC(P). That Tg(P) may be studied rather than TC(P) is consistent with the approximation that has been made previously [7] 1 ∂TC 1 ≈ TC ∂P Tg
Fig. 4. DDM predictions (curves based on Eq. (6)) for log(τ) vs. 1000/(TV γ) at various temperatures for PVAc and glycerol. For PVAc, γ = 2.55 and for glycerol, γ = 2.0.
∂Tg
∂Tg ∂P
! ð46Þ
The data of Reiser et al. [30] for meta-fluoroaniline and propylene carbonate were evaluated and the average values of φ were found to be 1.9 and 2.5 for, respectively, giving rise to values of φ + 1 of 2.9 and 3.5. These are to be compared with values of γ of 2.7 and 4.2 quoted by Reiser et al. [30]. Consequently, there appears to be a correlation between φ + 1 and γ. Because of the high value of γ = 8.5 that has been reported a chlorinated biphenyl (PCB62) is of interest [32]. Evaluation of the PVT data gives of φ ≈ 3.1 so that φ + 1 ≈ 4.1. This is the largest value of φ (and hence φ + 1) that is calculated so the correlation between φ + 1 and γ is consistent. However, 4.1 is much lower than the value of γ = 8.5 that has been reported. Consequently, it is concluded that φ is not equal to γ − 1 but rather is an important
J.T. Bendler et al. / Journal of Non-Crystalline Solids 357 (2011) 404–410
Fig. 6. Correlation length vs. temperature for glycerol. The points are data from Ref. [12] and the line is the DDM prediction.
factor in determining the value of γ. Again, this is not surprising considering Eq. (44). Clearly, the DDM gives insight into the origin of scaling, the most important factor being how TC depends on volume. This further shows why the scaling parameter is related to the ratio EV*/H*. Specifically, the chain rule and definition of the compressibility can be used to rewrite Eq. (12) as
α TV ∂TC
EV* H*
= DDM
1 + TP C
∂V T 1 + αP ðT−TC Þ
ð47Þ
An interesting form of Eq. (47) results when TC is a power law as in Eq. (43). In that case Eq. (47) reduces to the semi-quantitative equation
EV* 1−αP Tφ : ≈ H* DDM 1 + αP ðT−TC Þ
ð48Þ
Eq. (48) shows that in cases where TC is very volume sensitive (φ is large), EV* /H* is small. This makes physical sense because when the effect of volume is large EV* /H* is small. Finally, a non-scaling material, glycerol, is considered. Once again, the PVT data of Reiser et al. [30] are used. In order to generate the values of the relaxation time, Eq. (6) was best-fit to the data of Rossler [33] to obtain the zero pressure best-fit parameters. Finally, values of TC(P) were generated using the Tg(P) data of Reiser et al. [30] and Eq. (46) was applied. The results are shown in Fig. 4 for γ = 2.0 which is the value of γ that gave the closest family of curves. Values of γ = 1.3 [5] or 1.7 [30] gave curves that are much further apart. Careful inspection of Fig. 4 shows that the DDM results for glycerol do not fall
409
on a single curve. The curves are relatively close together but are not as close as those for PVAc. Interestingly, the lack of scaling is not due to a breakdown of the power law for TC. The data for glycerol, shown in Fig. 5, exhibit a power law with a value of φ ≈ 1.3. The value of φ is very small and thus TC is not strongly dependent upon volume. From Eq. (48), this is consistent with large values of EV*/H* and hence large effects of temperature relative to volume. However, glycerol does not scale as well as PVAc. In the DDM, it is this weak dependence of TC on volume and the factor of V in Eq. (42) that give rise to the breakdown of scaling in glycerol. The DDM also makes it possible to identify the physical reason for this behavior. The poor scaling for glycerol has been previously attributed to hydrogen bonding by Reiser et al. [30] and Roland et al. [5]. As has been discussed [7], candidates for defects in liquid glycerol are uncoordinated H-bonds, rather than vacancylike defects and such H-bond defects will not be much expanded over rigid, defect-free regions. It follows that defect pairing in glycerol will thus not be as readily influenced by pressure or volume as for a nonhydrogen bonded liquid. This gives rise to the weak dependence of TC on volume and hence the DDM explanation of poor scaling for glycerol.
5. Correlation lengths Additional insight into glass-forming liquids can be gained by considering correlation lengths. Considerable effort in recent years has focused on attempting to determine a dynamic heterogeneity length scale in glass-forming liquids. Listed in Table 1 are the experimental values for the dynamic heterogeneity length scale, ξ4D3CP, for four liquids measured using the 4D3CP NMR method pioneered by Tracht et al. [8–11]. These four liquids have been previously analyzed by Qui and Ediger [12]. The results for glycerol are also shown in Fig. 6. A correlation length, given by Eq. (8), is central to the DDM. Two quantities, TC and the short range length, L, are necessary for its evaluation at a given temperature. For each of the four materials studied by Qui and Ediger, the value of TC determined using dielectric relaxation data is listed in Table 1. In principle, L can be found from the → second moment of the direct correlation function, C r , between defects [34]. Alternatively, L may be treated as a phenomenological parameter and estimated by using Eq. (9). (Isotropy is assumed i.e. L1 = L2 = L3 = L.) That calculation requires another zero pressure model parameter, B*, in addition to c, do, and βKWW. The values of B* and βKWW were taken from the literature and are listed in Table 1. The values of do were approximated by the average distance between molecules and were calculated using the density and molecular weight. Since it is not known directly, the value of c was treated as an adjustable parameter. As will be seen, the choice of c has an independent implication that gives some justification for the choices that were made. The values of c and corresponding theoretical values of ξDDM are listed in Table 1. The theoretical values for glycerol are also
Table 1 ξTheory compared to 4D3CP NMR results ξExp DD 4D3CP for four liquids. Liquid T(K) Glycerol 199 Glycerol 203 Glycerol 207 PVAc 315 Sorbitol 275 OTP 252 OTP 252 a
ξExperiment nm 4D3CP a,b
1.3 ± 0.5 1.1 ± 0.5a,b 1.0 ± 0.5a,b 3.7 ± 1.0b,c 2.5 ± 1.2d 2.9 ± 1.0b 2.9 ± 1.0b
ξTheory DDM nm
βKWW
B*
TC K
c
L nm
do nm
Tg K
TB K
Log10Aτ
Log10τo
1.25 1.22 1.19 3.71 2.49 2.88 2.90
0.58e 0.58e 0.58e 0.53f 0.41g 0.52d 0.52d
18.9e 18.9e 18.9e 6.7i 5.2g 4.78j 16.4k
127.6e 127.6e 127.6e 250.3i 232 220.4j 181.6k
0.85 0.85 0.85 0.062 0.3 0.47 0.38
0.888 0.888 0.888 1.89 1.07 1.09 1.81
0.495 0.495 0.495 0.495 0.591 0.692 0.692
189e 189e 189e 302i 268l 246l 246l
285h 285h 285h ≈ 350i ≈ 335l ≈ 290m ≈ 290m
−15.16e −15.16e −15.16e −11.91i −13.4g −13.99j −19.3k
−15.3 −15.3 −15.3 −14.2 −14.7 −14.6 −20.1
Ref. [10]; bRef. [11]; cRefs. [8] and [9]; dRef. [12]; eRef. [15]; fRef. [15]; gC. Svanberg and R. Bergman, Phil. Mag. 87, 393 (2007). P. Lunkenheimer, U. Schneider, R. Brand, and A. Loidl, Contemp. Phys. 41, 15 (2000); iR. Richert, Physica A 287, 26 (2000); jRef. [35]; kRef. [37]; lRef. [36]. m C. Hansen, F. Stickel, T. Berger, R. Richert, E. W. Fischer, J. Chem. Phys. 107, 1086 (1997). h
410
J.T. Bendler et al. / Journal of Non-Crystalline Solids 357 (2011) 404–410
plotted in Fig. 6 where it is seen that the temperature variation of ξDDM and ξ4D3CP are similar. The values of c for glycerol, sorbitol, and OTP are relatively large and are on the order of that for equal occupancy of sites by defects and non-defects, c = 0.50. A smaller value of c (0.062) was necessary to obtain approximate agreement with experimental values of ξ4D3CP for PVAc. At the present time, experimental methods to independently determine c for a given liquid are uncertain. There is, however, supporting evidence for this difference. The pre-exponential for dielectric relaxation given by Eq. (6) contains both c and τo. Again, the approximation that δ = 0 was used i.e. the effects of thermal expansion and compressibility are ignored. As a consequence, values of the fitting parameters are slightly different than those quoted previously in this paper for which the effects of thermal expansion and compressibility are included. It is to be emphasized that, in the DDM, τo is not the preexponential itself, as it is usually defined in the standard Vogel relation, but rather it is the shortest time for a jump of a defect i.e. the pre-exponential, Aτ, is Aτ = c
−1 = β
τo :
ð49Þ
Physically, it is reasonable to expect a correspondence between the dynamical length ξ4D3CP measured by NMR and the defect–defect correlation length ξDDM. Certainly the size of the immobile regions should be directly related to ξDDM and the discussions of the NMR experiment in Refs. [8–10] indicate that ξ4D3CP is a measure of the diameter of the slow regions. Table 1 and Fig. 6 suggest a reasonable quantitative correspondence between ξDDM and ξ4D3CP. Acknowledgments This work was supported in part by the U.S. Office of Naval Research. JTB gratefully acknowledges financial support by the Department of Defense-Army Research Office (grant no. DAAD1901-1-0482). MFS would like to thank the Kinnear Chair of Physics for support. References [1] [2] [3] [4] [5]
The values of Aτ were used with Eq. (49) and the selected value of c to calculate the value of τo for each material. The results are listed in the last column of Table 1. It is noted that while the difference between the values of Aτ for glycerol and PVAc is more than three decades (log10(Aτ) = −15.16 for glycerol and −11.91 for PVAc), the values of τo differ by only about one decade in that log10(τo) = −15.3 for glycerol and −14.2 for PVAc. In addition, the value for sorbitol is about midway between. The similarity in values of τo and their closeness to lattice vibrational frequencies gives some justification to the choices of c. The relatively small values of both EV*/H* and c for PVAc can be understood in terms of polymer backbone constraints. The small value of EV*/H* shows that PVAc is a relatively volume sensitive material. To efficiently fill space and eliminate all “free volume” holes, the chain needs to reorganize itself, and this is much more difficult for the polymer than for most small molecule systems. This explains why most small molecule systems are less volume sensitive than PVAc. Again, because of its backbone constraints and the difficulty of reorganizing itself, PVAc is less defective than most small molecule systems. This explains the relatively small value of c observed for PVAc. The case of OTP is interesting. There is wide variation in values of the Vogel parameters. The largest difference in the values is between those for the data of Johari and Goldstein [35] and the more recent work [36,37]. For those parameters used in calculating the correlation length, the author's fit to Johari's data [35] yielded values of B* and TC, of 4.78 and 220.4 K respectively. A fit to the data provided by Naoki [37] yielded values of 16.38 and 181.61 K, respectively. More recent results by Wagner and Richert [36] are 10.83 and 168 K. However, as seen in Table 1, the values of c necessary to reproduce the correlation length of about 2.9 are relatively close to one another. The intriguing aspects of the OTP results are the pre-exponential, Aτ, and value of τo. The data of Johari and Goldstein result in a “normal” pre-exponential, log10(Aτ) =−13.99, and consequently a “normal” value of τo, log10(τo) =−14.6, the latter value being very close to the value for sorbitol. However, the data of Naoki et al. [37] (and Wagner and Richert [36] for which log10(Aτ) =−22.1) result in anomalously low values of on the order of 10−20 s. It will be of interest to investigate the cause of this disagreement.
[6]
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Contents lists available at ScienceDirect
Journal of Non-Crystalline Solids j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / j n o n c r y s o l
The defect diffusion model, scaling, EV*/H* and monomer volume and correlation lengths for glass-formers J.T. Bendler a, J.J. Fontanella b,⁎, M.F. Shlesinger b,c, M.C. Wintersgill b a b c
BSC, Inc., 3046 Player Drive, Rapid City, SD 57702, USA Physics Department, U.S. Naval Academy, Annapolis, MD 21402, USA Office of Naval Research, ONR Code 30, 875 N. Randolph St., Arlington VA 22203, USA
a r t i c l e
i n f o
Article history: Received 19 January 2010 Received in revised form 29 June 2010 Available online 21 August 2010 Keywords: Dielectric relaxation; High pressure; Scaling; Correlation length
a b s t r a c t Four topics are treated within the framework of the defect diffusion model (DDM). First, it is shown how the relationship between EV*/H* (ratio of the apparent isochoric activation energy to the isobaric activation enthalpy) and monomer volume for polymers that has been pointed out by Floudas and co-workers [G. Floudas, K. Mpoukouvalas and P. Papadopoulos, J. Chem. Phys. 124 (2006) 074905] is predicted. Next, it is shown that in the DDM, scaling arises because the critical temperature can be represented approximately by a power law. Consequently, in the DDM scaling is always approximate and significant departures from scaling, as is observed in the case of hydrogen bonded materials for example, are matters of degree. It is also shown how the connection of scaling with EV*/H* is a natural consequence of the DDM. Finally, DDM calculations of the defect correlation length are carried out and compared with experimental dynamical correlation lengths measured using the 4D3CP solid state NMR method. © 2010 Elsevier B.V. All rights reserved.
1. Introduction The effects of volume and temperature on the properties of glassforming liquids are topics of current interest [1–4]. In order to separate these effects, the ratio of the apparent isochoric activation energy, EV or EV*, EV = EV* =
∂ lnx ∂ð1=T Þ V
ð1Þ
to the isobaric activation enthalpy, EP or H* EP = H* =
∂ lnx ; ∂ð1=T Þ P
ð2Þ
where x is a transport property such as electrical relaxation time, viscosity or ionic conductivity, is often considered. This ratio can be written EV* = H*
∂ ln x ∂T V
=
∂ ln x : ∂T P
ð3Þ
Many experimental evaluations of this ratio have been reported [3,5] and the authors have recently derived an expression for the ratio ⁎ Corresponding author. Tel.: + 1 410 293 5507; fax: + 1 410 293 5508. E-mail addresses: [email protected] (J.T. Bendler), [email protected], [email protected] (J.J. Fontanella), [email protected] (M.F. Shlesinger), [email protected], [email protected] (M.C. Wintersgill). 0022-3093/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.jnoncrysol.2010.07.031
based on the defect diffusion model (DDM) [6,7]. In a recent paper, Floudas et al. have established a connection between EV*/H* and the monomer volume, Vm, for polymers [3]. They find that as Vm increases, in general EV*/H* decreases. This trend is accompanied by significant deviations. In the present paper, it is shown that this connection is accounted for by the DDM. A second experimental observation related to EV*/H* is known as scaling. Specifically, it has been found that for many glass-forming liquids ln(x) scales at least approximately with 1/(TVγ) and that scaling is related to EV*/H*. A review of the work until 2005 is given in Ref. [5]. In the present paper, it is shown how the phenomena related to scaling are natural consequences of the DDM. Finally, the DDM is used to calculate correlation lengths and the results are compared with experimental dynamical correlation lengths measured using the 4D3CP solid state NMR method [8–12]. 2. Theory The basic assumptions for the DDM are that a super-cooled liquid contains mobile single defects (MSDs), immobilized clustered single defects (ICSDs) and normal liquid molecules. In the case of fragile liquids, one may consider defects as persistent packets of free volume, MSDs being mobile defects with greater than average free volume and ICSDs having less than average free volume. Defects are persistent in that they are conserved, neither created nor destroyed, and simply converted thermodynamically between MSDs and ICSDs with changes in pressure and temperature. Transitory molecular configurations with instantaneous free volumes higher or lower than average are not permanent “defects” in this sense, but rather contribute to the high
J.T. Bendler et al. / Journal of Non-Crystalline Solids 357 (2011) 404–410
frequency background fluctuations of the molecular properties of the fluid. MSDs and ICSDs are associated with regions where the density is smaller and larger than average, respectively. Consequently, MSDs are large and ICSDs are small [13]. Because ICSD-rich regions of the fluid are rigid regions and MSD-rich regions are mobile, dynamic heterogeneity is an essential aspect of the DDM picture of super-cooled liquids. Finally, all of the physical properties of a fragile super-cooled liquid are influenced by the tendency of the MSDs to cluster (i.e., to form ICSDs/rigid regions) as temperature decreases or pressure increases. In the DDM the concentration of MSDs, c1, is given by [7] 0:5η
c1 = c exp −
βKWW B TC
!
ðT−TC Þ0:5η ð1−δÞ
:
ð4Þ
This leads to the following equation for the dielectric relaxation time −1 = β
τDDM = c
! 0:5η B TC Δo exp : kB T ðT−TC Þ0:5η ð1−δÞ
τo exp
ð5Þ
Δ0 is the smallest barrier opposing a defect hop. The exponential containing Δ0 together with the prefactor τo , give the shortest pausing time for the jump of a defect [14,15]. As usual, the Arrhenius term will be omitted so that the working equation is τDDM ≈c
−1 = β
B TC
0:5η
τo exp
!
ðT−TC Þ0:5η ð1−δÞ
:
ð6Þ
In Eqs. (5) and (6), c is the total concentration of defects (i.e., the fraction of lattice sites occupied by a defect), τo is the shortest time for the jump of a defect, and βKWW is the Kohlrausch–Williams–Watts or stretched exponential parameter associated with the relaxation function [16,17] β ϕðt Þ = A exp −ðt =τDDM Þ KWW :
ð7Þ
(1 − δ) = V(T,P)/Vo and V(T,P) is the volume of the liquid at pressure P, and absolute temperature T, and Vo is a reference volume at P = 0 at a given reference temperature. For consistency with the notation given in other papers in the literature, for the remainder of this paper, V(T,P)/ Vo will be referred to simply as V. η is an integer (1, 2 or 3) describing the dimensionality of the correlation volume [18]. If δ = 0 and η = 3, a 3/2 power law results. Materials such as poly(propylene glycol) are successfully described by η = 3 [13]. If, on the other hand, the correlation volume grows in a two-dimensional manner, with η = 2, then Eq. (6) reduces to the standard Vogel equation [19], (assuming that volume is not strongly dependent upon pressure or temperature i.e. δ = 0). For example, glycerol is best described by η = 2 [13]. A correlation length, ξj,DDM, corresponding roughly to the distance beyond which the defects do not interact, is given by [14,18] ξj;DDM ðT Þ = Lj
TC T−TC
0:5
:
ð8Þ
TC is the critical temperature at which all defects would be clustered ICSDs if the glass transition did not intervene. The critical temperature, TC, depends on pressure, i.e. TC = TC(P). (See also Eq. (13) below where a simple Bragg–Williams lattice model for TC is discussed.) The correlation length of Eq. (8) plays a central role in the DDM analysis, and, in particular, in the derivation of the VTF-like temperature laws such as Eq. (6) [20,21]. As will be shown, this quantity appears to be associated with experimental dynamical correlation lengths. As the temperature falls, more defects become immobile, locked into clusters and thus unavailable to promote relaxation. Only mobile single (uncorrelated) defects (MSDs) remain effective in producing relaxa-
405
tion, and this depletion of MSDs with falling temperature leads directly to Eq. (6). Another parameter in the exponent of Eq. (6) is
B =−
L1 L2 L3 lnð1−cÞ : d3o βKWW
ð9Þ
do is the average distance between neighboring sites at P = 0 and at a given reference temperature. The three indices on the “direct” correlation lengths Lj allow for anisotropic defect–defect interactions. If the interactions are isotropic, then L1 = L2 = L3 = L. As has been pointed out [7], Eq. (6) is only applicable between TB (or TLL) and Tg. It is not valid below Tg because rigidity percolates below Tg and the motion of the MSDs is thereby restricted. In addition, Eq. (6) is not expected to apply to temperatures above TB (or TLL) which is on the order of 1.2 Tg. For temperatures above TB, we assume that jamming caused by the ICSD-rich regions is greatly diminished and sufficient free volume becomes available to allow new relaxation processes in addition to those controlled by MSD hopping diffusion. Consequently, the properties of the liquid will vary differently with temperature and pressure above and below TB but the changes in behavior are likely to be continuous and small. In fact, property changes near TB are often subtle, being observable primarily via the temperature dependence of a property rather than as a discontinuous change in a property itself [22]. As shown elsewhere [7], if the isothermal compressibility, κT, and isobaric volume thermal expansion coefficient, αP, are defined in the usual manner, κT = −
1 ∂V V ∂P T
ð10Þ
and 1 ∂V ; V ∂T P
αP =
ð11Þ
the assumption that TC = TC(P) leads to
EV* H*
αP T ∂TC κT TC ∂P T = : α ðT−TC Þ 1+ P 0:5η 1−
DDM
ð12Þ
(An approximate version of Eq. (12) was given in an earlier note [6].) Eq. (12) has been used to predict the ratio EV*/H* and its temperature dependence for several materials [7] and good agreement between theory and experiment is found. An interesting form of Eq. (12) is obtained using a simple Bragg– Williams treatment of a (defect) phase separation transition (for nearestneighbor pair interactions with equal occupancy of A and B sites). It has been pointed out that the critical demixing temperature is given by [23] TC =
zjΔhj zðjΔε + PΔvjÞ = 4kB 4kB
ð13Þ
where z is the lattice coordination number, kB is Boltzmann's constant, and Δh = Δε + PΔv is the decrease in enthalpy resulting from the formation of a defect pair: defect
+
defect ↔
ðdefectÞ2
ð14Þ
where Δε is the decrease in pair energy and Δv is the decrease in volume. From Eq. (13) one finds ð∂TC =∂P ÞT ≈zjΔvj = 4kB
ð15Þ
406
J.T. Bendler et al. / Journal of Non-Crystalline Solids 357 (2011) 404–410
so that the controlling molecular parameter is the net (negative) volume change, Δv, caused by defect pairing. If Δv is negative (as we assume for a fragile liquid), then a pressure increase (or volume decrease) causes mobile single defects (MSDs) to cluster (and rigidity to increase) as the equilibrium in Eq. (14) shifts to the right. Substituting Eqs. (13) and (15) into Eq. (12) gives
α T Δv 1− κP T Δh
EV* H*
= DDM
1+
αP ðT−TC Þ 0:5η
:
ð16Þ
This shows that the main factor determining the value of EV*/H* is the ratio of the volume change to the change in enthalpy when defects cluster. 3. Correlation of E*V/H* with monomer volume In this section it will be shown that Eq. (12) implies that for polymers, as the monomer volume, Vm, increases, EV*/H* decreases. For simplicity, it will be assumed that η = 2 and this (Vogel) approximation give rise to the following equation.
EV* H*
α T Δv 1− κP T Δh T C ∂P = = 1 + αP ðT−TC Þ 1 + αP ðT−TC Þ α ∂T 1− κ P TT C
DDM
ð17Þ
We first focus on Δν. If it is assumed that the rigid regions are “crystal-like” and the mobile regions are “liquid-like,” Δν can be estimated from the Clapyron equation:
dP dT
= coexist
ΔSmelt ΔVmelt
ð18Þ
First, the total volume of the material is written as VT = Nm Vm + Vf + Vdefect :
ð23Þ
Nm is the number of monomers, Vm is the volume of a monomer, Vf is the free volume not associated with defects and Vdefect is the total volume of the defects. Defining V IC to be the average volume of an ICSD and V MS to be the average volume of an MSD, it follows that Vdefect = NIC V IC + NMS V MS
ð24Þ
where NIC is the number of ICSDs, and NMS is the number of MSDs. Next, if the simplest case is assumed where two MSDs form a single ICSD Δv = 2 V MS −V IC :
ð25Þ
Written in this manner, Δv is a positive value. The thermal expansion coefficient can be calculated from αP =
1 ∂ Nm Vm + Vf + NIC V IC + NMS V MS VT ∂T
ð26Þ
Next, it is assumed that the most temperature dependent terms in Eq. (26) are NIC and NMS. This is a reasonable assumption because Nm is not temperature dependent. The remaining volumes, Vm, Vf, V IC and V MS are somewhat temperature dependent, but the variation with temperature is expected to be small compared with that for the numbers of defects. Consequently, αP =
1 VT
∂N ∂NIC + V MS MS V IC ∂T ∂T
ð27Þ
Again assuming that two MSDs form one ICSD, where the quantity on the left is the slope of the solid–liquid coexistence line, ΔSmelt is the entropy change on melting and ΔVmelt is the volume change on melting. If the heat absorbed on melting (or release on freezing) is measured, then ΔSmelt can be determined from ΔSmelt =
ΔHmelt Tmelt
ð19Þ
−2
Δv≈ΔVmelt
αP =
∂NIC 1 Δv ∂NIC =− V IC −2V MS VT VT ∂T ∂T
:
ð29Þ
Alternatively, αP can be written ð20Þ
As a consequence, Δν can be estimated in several ways: (a) ΔVmelt can be determined directly from the density of the liquid and the density of the crystalline solid along with use of the approximation Δν≈ ΔVmelt; (b) ΔVmelt can be determined directly from Eqs. (18) and (19)) along with use of the approximation Δν ≈ ΔVmelt; (c) If the slope of the crystal–liquid line, (dTm/dP) is not known, then the approximation 1 dTg 1 dTmelt ≈ Tg dP Tmelt dP
ð28Þ
Consequently, the thermal expansion coefficient is
Finally, if it is assumed that Δν ≈ ΔVmelt, ΔHmelt dTmelt = : Tmelt dP coexist
∂NIC ∂NMS = : ∂T ∂T
αP = +
1 Δv ∂NMS 2 VT ∂T
ð30Þ
The numbers of defects can be written in terms of the defect concentrations as [13] NIC =
N ðc−c1 Þ 2
ð31Þ
or ð21Þ
NMS = Nc1
ð32Þ
can be used along with Eq. (20). The validity of the approximation represented by Eq. (21) has been discussed elsewhere [15,24]; (d) In addition, the following approximation
where N is the number of lattice sites. Consequently, the thermal expansion coefficient can be rewritten as
4k dTC 4k T dTg ≈ B C Δv≈ B z dP z Tg dP
αP = +
ð22Þ
could be used where z has a value from 8 to 12; (e) For the present work, the following approach based on the thermal expansion coefficient is taken.
N Δv ∂c1 2 VT ∂T
ð33Þ
(This follows from either Eq. (29) or (30).) Next, using the assumptions that lead to the Vogel equation, Eq. (4) becomes c1 = c expð−βKWW B*TC = ðT−TC ÞÞ
ð34Þ
J.T. Bendler et al. / Journal of Non-Crystalline Solids 357 (2011) 404–410
407
so that ∂c1 β B*T = c KWW 2C expð−βKWW B*TC = ðT−TC ÞÞ ∂T ðT−TC Þ
ð35Þ
This gives us the following equation for Δν Δv =
2αP VT Nc
βKWW B*TC ðT−TC Þ2
expð−βKWW B*TC = ðT−TC ÞÞ
ð36Þ
Next, because most of the material is made up of molecules, it follows from Eq. (23) that V T e Nm V m :
ð37Þ
Also, the usual definition of the lattice (one lattice site per molecule) is used so that N ~ Nm Δv =
2αP Vm c
βKWW B*TC ðT−TC Þ2
expð−βKWW B*TC = ðT−TC ÞÞ:
ð38Þ
Finally, then, Eq. (17) can be rewritten as
EV* H*
= DDM
2 ðT−TC Þ2 Vm T P 1−2α κT jΔhjTC cβKWW B* expð−βKWW B*TC = ðT−TC ÞÞ
1−αP ðT−TC Þ
ð39Þ
This shows that, if all other parameters were the same for different materials, as the monomer volume increases, the ratio decreases. That is consistent with the empirical observation of Floudas et al. [3] that there is a general tendency for the ratio to decrease as monomer volume increases. However, it is also clear that factors other than Vm also affect EV*/H* in that a plot of EV*/H* vs. Vm does not form a single curve [3]. Eq. (39) also predicts this result because different materials exhibit different values of the characteristic parameters, c, B, βKWW, etc. 4. Scaling
Fig. 2. Experimental results of Heinrich and Stoll [27] (points) and DDM predictions (six theoretical curves based on Eq. (6)) for log(τ) vs. specific volume at various temperatures for PVAc. The symbols are: Solid triangles—393 K; Open triangles—383 K; Solid circles— 373 K; Open circles—363 K; Solid squares—353 K; Open squares—343 K.
Stoll [27], shown in Figs. 1 and 2 for PVAc at 393 K and below, were analyzed. The reasons for limiting the temperature range of the data are discussed elsewhere [7]. The authors have recently considered that data using the 3/2 power (η = 3) form of Eq. (6) [7]. For comparison and more relevance to the literature, the data were reconsidered using the DDM Vogel-form (η = 2) of Eq. (6). The values of V were determined using the Tait equation in the form dðT Þ + P V = eðT Þ 1−g ðT Þ ln dðT Þ + Po
ð40Þ
The values of the fitting parameters, d(T), e(T), and g(T) are given in Ref. [7]. In order to represent TC(P), the following empirical relation was used bP 1 = b TC ðP Þ = TC ð0Þ 1 + a
ð41Þ
A concept related to the ratio EV*/H* is scaling [5,25,26]. It has been suggested that when the relaxation time is plotted vs. TVγ, the constant (scaling parameter) γ can be adjusted so that the data form a single curve, hence the term scaling. In order to evaluate the concept of scaling within the framework of the DDM, the data of Heinrich and
According to Mpoukouvalas et al. [28], an equation analogous to Eq. (41) was first introduced by Simon and Glatzel [29]. The quantities a, b and TC(0) are fitting constants. The best-fit parameters were found
Fig. 1. Experimental results of Heinrich and Stoll [27] (points) and DDM predictions (six theoretical curves based on Eq. (6)) for log(τ) vs. pressure at various temperatures for PVAc. The symbols are: Solid triangles—393 K; Open triangles—383 K; Solid circles— 373 K; Open circles—363 K; Solid squares—353 K; Open squares—343 K.
Fig. 3. Experimental results of Heinrich and Stoll [27] (points) and DDM predictions (six theoretical curves based on Eq. (6)) for log(τ) vs. 1000/(TV2.5) at various temperatures for PVAc. The symbols are: Solid triangles—393 K; Open triangles—383 K; Solid circles— 373 K; Open circles—363 K; Solid squares—353 K; Open squares—343 K.
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J.T. Bendler et al. / Journal of Non-Crystalline Solids 357 (2011) 404–410
to be log10(Aτ) = −11.17, B* = 6.06, TC(0) = 261.5 K, and a = 1.33 GPa and b = 4.47. Figs. 1 and 2 are plots of log(τ) vs. P and log(τ) vs. V/Vo, respectively. In both figures, both the data and best-fit DDM theory are shown. Clearly, it is possible to represent all of the pressure and temperature data using the DDM. Fig. 2 makes it clear that the DDM is capable of explaining the effects of temperature at a constant volume even though it is a free volume theory. Fig. 3 is a scaling plot of the same data and theory where γ = 2.55. The data for PVAc appear to scale and the value of the scaling parameter is in agreement with Roland et al. who quote a value for PVAc of γ = 2.6 [5]. The important feature is that log(τ) for PVAc calculated using the DDM appears to scale. However, a close inspection of Fig. 3 shows that the scaling predicted by the DDM is not perfect. This is more apparent in Fig. 4 where the predictions of the DDM only (for six isothermal curves) are plotted. It is clear that there is a slight scatter in the prediction and the scatter is on the order of the width of the lines. It was not possible to achieve better scaling by adjusting γ. This shows that for PVAc, at least, scaling is an approximate concept based on the DDM. Nonetheless, the predictions of the DDM make it possible to gain some insight into the reason for scaling. Toward this end, Eq. (6) is rewritten as follows. 0
1
B 0:434B* C log10 ðτDDM Þ≈ log10 Aτ + @ A T T ðP Þ−1 V
ð42Þ
C
where Vogel behavior (η = 2) has once again been assumed. Since at least approximate scaling exists in the DDM, the behavior must be caused by the denominator of Eq. (42). In fact, in the DDM, scaling arises because in the experimental range of volumes and temperatures TC ðP Þ≈kV
−φ
ð43Þ
where k is a constant. To show that this is true, TC(P) for PVAc (Eq. (10)) was plotted vs. V using the PVT data given by Eq. (40). The results are shown in Fig. 5 for 393 K. The best-fit values of k and φ are 284 K and 1.52, respectively. Further, the parameters k and φ are only weakly temperature dependent, varying from about 284 K to 272 K and 1.52 to 1.57, respectively, over the temperature range 393 K to 343 K. Consequently, Eq. (42) can be rewritten log10 ðτDDM Þ≈ log10 Aτ +
0:434B*k TV φ + 1 −kV
ð44Þ
Fig. 5. Tα vs. V for PVAc (393 K), glycerol (280 K), meta-fluoroaniline (280 K), and propylene carbonate (280 K) along with a plot of TC vs. V for PVAc at 393 K. Also included are the best-fit power law curves (Eq. (43)).
This shows that the DDM equation for the relaxation time has the functional dependence necessary for scaling, namely τDDM = F (TVpower) where F is a function. Interestingly, for PVAc, the fit gives φ ≈ 1.5 and γ ≈ 2.55. Consequently, based on the numbers for PVAc, it is tempting to conclude that the exponent in Eq. (44) is γ, i.e. it is tempting to conclude that γ = φ + 1. However, the DDM does not suggest that. Eq. (44) shows an extra factor of kV in the denominator and V, itself, is temperature dependent. Consequently, it would not be expected that γ = φ + 1 in general. To check this result, other materials were evaluated. Two other materials that exhibit scaling, meta-fluoroaniline and propylene carbonate, were studied by Reiser et al [30]. However, Reiser et al. did not report numbers for relaxation times so TC(P) could not be determined. However, Tg(P) was reported, albeit in a slightly different form
Tg ðP Þ ≡ Tg ðP = 0Þ +
∂P
P+
2 1 ∂ Tg 2 P : 2 ∂P 2
ð45Þ
Consequently, power law behavior was studied for Tg(P) (or, more precisely, Tα(P)) rather than TC(P). In order to verify that this makes little difference, Tg(P) for PVAc from Roland and Casalini [31] was plotted vs. V and the results are also shown in Fig. 5. The value of φ was 1.64 in reasonable agreement with the value of 1.52 determined from TC(P). A value of k = 340 K was also obtained showing that Tg(P) is larger than TC(P). That Tg(P) may be studied rather than TC(P) is consistent with the approximation that has been made previously [7] 1 ∂TC 1 ≈ TC ∂P Tg
Fig. 4. DDM predictions (curves based on Eq. (6)) for log(τ) vs. 1000/(TV γ) at various temperatures for PVAc and glycerol. For PVAc, γ = 2.55 and for glycerol, γ = 2.0.
∂Tg
∂Tg ∂P
! ð46Þ
The data of Reiser et al. [30] for meta-fluoroaniline and propylene carbonate were evaluated and the average values of φ were found to be 1.9 and 2.5 for, respectively, giving rise to values of φ + 1 of 2.9 and 3.5. These are to be compared with values of γ of 2.7 and 4.2 quoted by Reiser et al. [30]. Consequently, there appears to be a correlation between φ + 1 and γ. Because of the high value of γ = 8.5 that has been reported a chlorinated biphenyl (PCB62) is of interest [32]. Evaluation of the PVT data gives of φ ≈ 3.1 so that φ + 1 ≈ 4.1. This is the largest value of φ (and hence φ + 1) that is calculated so the correlation between φ + 1 and γ is consistent. However, 4.1 is much lower than the value of γ = 8.5 that has been reported. Consequently, it is concluded that φ is not equal to γ − 1 but rather is an important
J.T. Bendler et al. / Journal of Non-Crystalline Solids 357 (2011) 404–410
Fig. 6. Correlation length vs. temperature for glycerol. The points are data from Ref. [12] and the line is the DDM prediction.
factor in determining the value of γ. Again, this is not surprising considering Eq. (44). Clearly, the DDM gives insight into the origin of scaling, the most important factor being how TC depends on volume. This further shows why the scaling parameter is related to the ratio EV*/H*. Specifically, the chain rule and definition of the compressibility can be used to rewrite Eq. (12) as
α TV ∂TC
EV* H*
= DDM
1 + TP C
∂V T 1 + αP ðT−TC Þ
ð47Þ
An interesting form of Eq. (47) results when TC is a power law as in Eq. (43). In that case Eq. (47) reduces to the semi-quantitative equation
EV* 1−αP Tφ : ≈ H* DDM 1 + αP ðT−TC Þ
ð48Þ
Eq. (48) shows that in cases where TC is very volume sensitive (φ is large), EV* /H* is small. This makes physical sense because when the effect of volume is large EV* /H* is small. Finally, a non-scaling material, glycerol, is considered. Once again, the PVT data of Reiser et al. [30] are used. In order to generate the values of the relaxation time, Eq. (6) was best-fit to the data of Rossler [33] to obtain the zero pressure best-fit parameters. Finally, values of TC(P) were generated using the Tg(P) data of Reiser et al. [30] and Eq. (46) was applied. The results are shown in Fig. 4 for γ = 2.0 which is the value of γ that gave the closest family of curves. Values of γ = 1.3 [5] or 1.7 [30] gave curves that are much further apart. Careful inspection of Fig. 4 shows that the DDM results for glycerol do not fall
409
on a single curve. The curves are relatively close together but are not as close as those for PVAc. Interestingly, the lack of scaling is not due to a breakdown of the power law for TC. The data for glycerol, shown in Fig. 5, exhibit a power law with a value of φ ≈ 1.3. The value of φ is very small and thus TC is not strongly dependent upon volume. From Eq. (48), this is consistent with large values of EV*/H* and hence large effects of temperature relative to volume. However, glycerol does not scale as well as PVAc. In the DDM, it is this weak dependence of TC on volume and the factor of V in Eq. (42) that give rise to the breakdown of scaling in glycerol. The DDM also makes it possible to identify the physical reason for this behavior. The poor scaling for glycerol has been previously attributed to hydrogen bonding by Reiser et al. [30] and Roland et al. [5]. As has been discussed [7], candidates for defects in liquid glycerol are uncoordinated H-bonds, rather than vacancylike defects and such H-bond defects will not be much expanded over rigid, defect-free regions. It follows that defect pairing in glycerol will thus not be as readily influenced by pressure or volume as for a nonhydrogen bonded liquid. This gives rise to the weak dependence of TC on volume and hence the DDM explanation of poor scaling for glycerol.
5. Correlation lengths Additional insight into glass-forming liquids can be gained by considering correlation lengths. Considerable effort in recent years has focused on attempting to determine a dynamic heterogeneity length scale in glass-forming liquids. Listed in Table 1 are the experimental values for the dynamic heterogeneity length scale, ξ4D3CP, for four liquids measured using the 4D3CP NMR method pioneered by Tracht et al. [8–11]. These four liquids have been previously analyzed by Qui and Ediger [12]. The results for glycerol are also shown in Fig. 6. A correlation length, given by Eq. (8), is central to the DDM. Two quantities, TC and the short range length, L, are necessary for its evaluation at a given temperature. For each of the four materials studied by Qui and Ediger, the value of TC determined using dielectric relaxation data is listed in Table 1. In principle, L can be found from the → second moment of the direct correlation function, C r , between defects [34]. Alternatively, L may be treated as a phenomenological parameter and estimated by using Eq. (9). (Isotropy is assumed i.e. L1 = L2 = L3 = L.) That calculation requires another zero pressure model parameter, B*, in addition to c, do, and βKWW. The values of B* and βKWW were taken from the literature and are listed in Table 1. The values of do were approximated by the average distance between molecules and were calculated using the density and molecular weight. Since it is not known directly, the value of c was treated as an adjustable parameter. As will be seen, the choice of c has an independent implication that gives some justification for the choices that were made. The values of c and corresponding theoretical values of ξDDM are listed in Table 1. The theoretical values for glycerol are also
Table 1 ξTheory compared to 4D3CP NMR results ξExp DD 4D3CP for four liquids. Liquid T(K) Glycerol 199 Glycerol 203 Glycerol 207 PVAc 315 Sorbitol 275 OTP 252 OTP 252 a
ξExperiment nm 4D3CP a,b
1.3 ± 0.5 1.1 ± 0.5a,b 1.0 ± 0.5a,b 3.7 ± 1.0b,c 2.5 ± 1.2d 2.9 ± 1.0b 2.9 ± 1.0b
ξTheory DDM nm
βKWW
B*
TC K
c
L nm
do nm
Tg K
TB K
Log10Aτ
Log10τo
1.25 1.22 1.19 3.71 2.49 2.88 2.90
0.58e 0.58e 0.58e 0.53f 0.41g 0.52d 0.52d
18.9e 18.9e 18.9e 6.7i 5.2g 4.78j 16.4k
127.6e 127.6e 127.6e 250.3i 232 220.4j 181.6k
0.85 0.85 0.85 0.062 0.3 0.47 0.38
0.888 0.888 0.888 1.89 1.07 1.09 1.81
0.495 0.495 0.495 0.495 0.591 0.692 0.692
189e 189e 189e 302i 268l 246l 246l
285h 285h 285h ≈ 350i ≈ 335l ≈ 290m ≈ 290m
−15.16e −15.16e −15.16e −11.91i −13.4g −13.99j −19.3k
−15.3 −15.3 −15.3 −14.2 −14.7 −14.6 −20.1
Ref. [10]; bRef. [11]; cRefs. [8] and [9]; dRef. [12]; eRef. [15]; fRef. [15]; gC. Svanberg and R. Bergman, Phil. Mag. 87, 393 (2007). P. Lunkenheimer, U. Schneider, R. Brand, and A. Loidl, Contemp. Phys. 41, 15 (2000); iR. Richert, Physica A 287, 26 (2000); jRef. [35]; kRef. [37]; lRef. [36]. m C. Hansen, F. Stickel, T. Berger, R. Richert, E. W. Fischer, J. Chem. Phys. 107, 1086 (1997). h
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plotted in Fig. 6 where it is seen that the temperature variation of ξDDM and ξ4D3CP are similar. The values of c for glycerol, sorbitol, and OTP are relatively large and are on the order of that for equal occupancy of sites by defects and non-defects, c = 0.50. A smaller value of c (0.062) was necessary to obtain approximate agreement with experimental values of ξ4D3CP for PVAc. At the present time, experimental methods to independently determine c for a given liquid are uncertain. There is, however, supporting evidence for this difference. The pre-exponential for dielectric relaxation given by Eq. (6) contains both c and τo. Again, the approximation that δ = 0 was used i.e. the effects of thermal expansion and compressibility are ignored. As a consequence, values of the fitting parameters are slightly different than those quoted previously in this paper for which the effects of thermal expansion and compressibility are included. It is to be emphasized that, in the DDM, τo is not the preexponential itself, as it is usually defined in the standard Vogel relation, but rather it is the shortest time for a jump of a defect i.e. the pre-exponential, Aτ, is Aτ = c
−1 = β
τo :
ð49Þ
Physically, it is reasonable to expect a correspondence between the dynamical length ξ4D3CP measured by NMR and the defect–defect correlation length ξDDM. Certainly the size of the immobile regions should be directly related to ξDDM and the discussions of the NMR experiment in Refs. [8–10] indicate that ξ4D3CP is a measure of the diameter of the slow regions. Table 1 and Fig. 6 suggest a reasonable quantitative correspondence between ξDDM and ξ4D3CP. Acknowledgments This work was supported in part by the U.S. Office of Naval Research. JTB gratefully acknowledges financial support by the Department of Defense-Army Research Office (grant no. DAAD1901-1-0482). MFS would like to thank the Kinnear Chair of Physics for support. References [1] [2] [3] [4] [5]
The values of Aτ were used with Eq. (49) and the selected value of c to calculate the value of τo for each material. The results are listed in the last column of Table 1. It is noted that while the difference between the values of Aτ for glycerol and PVAc is more than three decades (log10(Aτ) = −15.16 for glycerol and −11.91 for PVAc), the values of τo differ by only about one decade in that log10(τo) = −15.3 for glycerol and −14.2 for PVAc. In addition, the value for sorbitol is about midway between. The similarity in values of τo and their closeness to lattice vibrational frequencies gives some justification to the choices of c. The relatively small values of both EV*/H* and c for PVAc can be understood in terms of polymer backbone constraints. The small value of EV*/H* shows that PVAc is a relatively volume sensitive material. To efficiently fill space and eliminate all “free volume” holes, the chain needs to reorganize itself, and this is much more difficult for the polymer than for most small molecule systems. This explains why most small molecule systems are less volume sensitive than PVAc. Again, because of its backbone constraints and the difficulty of reorganizing itself, PVAc is less defective than most small molecule systems. This explains the relatively small value of c observed for PVAc. The case of OTP is interesting. There is wide variation in values of the Vogel parameters. The largest difference in the values is between those for the data of Johari and Goldstein [35] and the more recent work [36,37]. For those parameters used in calculating the correlation length, the author's fit to Johari's data [35] yielded values of B* and TC, of 4.78 and 220.4 K respectively. A fit to the data provided by Naoki [37] yielded values of 16.38 and 181.61 K, respectively. More recent results by Wagner and Richert [36] are 10.83 and 168 K. However, as seen in Table 1, the values of c necessary to reproduce the correlation length of about 2.9 are relatively close to one another. The intriguing aspects of the OTP results are the pre-exponential, Aτ, and value of τo. The data of Johari and Goldstein result in a “normal” pre-exponential, log10(Aτ) =−13.99, and consequently a “normal” value of τo, log10(τo) =−14.6, the latter value being very close to the value for sorbitol. However, the data of Naoki et al. [37] (and Wagner and Richert [36] for which log10(Aτ) =−22.1) result in anomalously low values of on the order of 10−20 s. It will be of interest to investigate the cause of this disagreement.
[6]
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