Temperature dependent micro-rheology of a glass-forming polymer melt studied by molecular dynamics simulation A. Kuhnhold and W. Paul Citation: The Journal of Chemical Physics 141, 124907 (2014); doi: 10.1063/1.4896151 View online: http://dx.doi.org/10.1063/1.4896151 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/12?ver=pdfcov Published by the AIP Publishing Articles you may be interested in On the Bauschinger effect in supercooled melts under shear: Results from mode coupling theory and molecular dynamics simulations J. Chem. Phys. 138, 12A513 (2013); 10.1063/1.4770336 Quantifying chain reptation in entangled polymer melts: Topological and dynamical mapping of atomistic simulation results onto the tube model J. Chem. Phys. 132, 124904 (2010); 10.1063/1.3361674 Probe molecules in polymer melts near the glass transition: A molecular dynamics study of chain length effects J. Chem. Phys. 132, 034901 (2010); 10.1063/1.3284780 The measurement of mechanical properties of glycerol, m -toluidine, and sucrose benzoate under consideration of corrected rheometer compliance: An in-depth study and review J. Chem. Phys. 129, 074502 (2008); 10.1063/1.2965528 A cryostat and temperature control system optimized for measuring relaxations of glass-forming liquids Rev. Sci. Instrum. 79, 045105 (2008); 10.1063/1.2903419
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THE JOURNAL OF CHEMICAL PHYSICS 141, 124907 (2014)
Temperature dependent micro-rheology of a glass-forming polymer melt studied by molecular dynamics simulation A. Kuhnholda) and W. Paul Institut für Physik, Martin-Luther-Universität Halle-Wittenberg, 06099 Halle (Saale), Germany
(Received 21 May 2014; accepted 9 September 2014; published online 26 September 2014) We present a Molecular Dynamics simulation study of a micro-rheological probing of the glass transition in a polymer melt. Our model system consists of short bead-spring chains and the temperature ranges from well above the glass transition temperature to about 10% above it. The nano-particle clearly couples to the slowing down of the polymer segments and the calculated storage and loss moduli reveal the approach to the glass transition. At temperatures close to the mode coupling Tc of the polymer melt, the micro-rheological moduli measure the local viscoelastic response of the cage of monomers surrounding the nano-particle and no longer reveal the true melt moduli. The incoherent scattering function of the nano-particle exhibits a stretched exponential decay, typical for the α-process in glass forming systems. We find no indication of a strong superdiffusive regime as has been deduced from a recent experiment in the same temperature range but for smaller momentum transfers. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4896151] I. INTRODUCTION
Passive micro-rheology has become the method of choice for studying linear viscoelastic properties of soft matter especially for small and biological samples, because of the possibility to receive information in a non-invasive/non-destructive manner, by studying the Brownian motion of embedded (sub-)micron sized particles. This motion is connected to the complex modulus of the sample in a wide frequency regime. The underlying theories for the analysis of the Brownian motion range from a rather simple Generalized Stokes-Einstein relation1–3 to full hydrodynamic approaches.4–9 When approaching the glass transition temperature, the mobility of polymeric liquids decreases by orders of magnitude. The glass transition itself is still not fully understood and has found much theoretical and experimental interest.10–14 The glass transition of a pure polymeric system of a model similar to that used here has already been studied in great detail.14, 15 The interesting point is now to see how dispersed nano-particles couple to the decreasing mobility of the polymer chains, which is theoretically described by the so-called caging effect. Many experiments in this field are done by optical measurements, e.g., diffusing wave spectroscopy (DWS) and dynamic light scattering (DLS), where the quantity of interest is the incoherent intermediate scattering function of dispersed micron sized particles. Recent studies with suspended gold nano-particles measured by X-ray photon correlation spectroscopy (XPCS) find a compressed exponential behavior of the incoherent intermediate scattering function with exponent larger than 1, which would indicate a superdiffusive motion of the nano-particles at long times.16, 17 By using a simple Gaussian approximation to obtain the mean squared displacement from the scattering function, they also look at rheological properties such as the temperature dependent viscosity, which is a good predictor for the glass transition. a) Electronic mail: [email protected]
0021-9606/2014/141(12)/124907/10/$30.00
In a previous publication,18 we showed for the same model as is studied in this work that the micro-rheological approach does work very well over a wide frequency range if the particle-monomer-interaction is not too strong. We carefully examined the effect of analyzing the particle mean squared displacement on several levels of theoretical approximation, ranging from the original Stokes-Einstein approach suggested by Mason and Weitz,1 to a complete consideration of hydrodynamic effects including fluid inertia and vortex generation by the particle motion. Including the latter two effects proved to be necessary for strong attractive interactions between the nano-particle and the monomers to avoid unphysical behavior of the obtained moduli. For relatively weak attraction the original Stokes-Einstein level description including the inertia term for the nano-particle provided an almost quantitative estimate for the melt moduli. For this reason we chose here a rather weak attraction between nano-particle and polymer melt to study the influence of lowering the temperature on the results of such micro-rheological measurements. We will first introduce the model that we use for our simulation study in Sec. II. After this we will show the determination of the temperature dependent melt modulus following the method introduced in Ref. 19 in Sec. III. Section IV then presents the micro-rheological approach to obtain the melt modulus from the Brownian motion of the suspended nano-particle. To compare our results to recent experiment we also look at the incoherent intermediate scattering function of the nano-particle in this section. Finally, Sec. V will offer our conclusions.
II. MODEL AND SIMULATION
For the simulations, we use a bead-spring polymer model, that is similar to that of Kremer and Grest,20 where all pairs of monomers interact via a truncated and shifted LJ and where the bonds are Lennard-Jones potential Umm
141, 124907-1
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J. Chem. Phys. 141, 124907 (2014)
included by a FENE potential UF with a spring constant k = 30 and a maximum elongation RF = 1.5σ mm , where σ mm is the diameter of a bead: 12 6 σmm σ 4 + Ush (r), r ≤ rc1 − mm mm LJ r r , Umm (r) = 0, r > rc1 (1)
k 2 r 2 UF (r) = − RF ln 1 − . 2 RF
(2)
To have vanishing potential, force and first derivative of force at the cut-off distance rc1 , the shift function Ush is added. The form and derivation of this function can be found in the manual of the GROMACS-4.5.4 Molecular Dynamics simulation package, which is used for the simulations.21, 22 In the original Kremer-Grest model only the repulsive part of the LennardJones potential is used by setting the cut-off distance to rc1 = 21/6 σ mm . For the investigation of temperature dependent behavior, it is mandatory to use also the attractive part of the potential by setting rc1 = 2 · 21/6 σ mm , so that the model is no longer athermal.23 In the following we use reduced LennardJones units for all numerical values, i.e., mm = 1, σ mm = 1, σ m = 1, τ = √ mm = 1. For studying the micro-rheology of mm /m
our system, we add a nano-particle, which is modeled as a hard sphere of radius R0 , that interacts with the monomers by a Lennard-Jones potential similar to that in Eq. (1), but with a hard sphere shift in the denominator: ⎧ σ 6 σpm 12 ⎨ pm +Ush (r), r ≤ rc2 4pm r−R − r−R LJ (r) = . Upm 0 0 ⎩ 0, r > rc2 (3) To have also the attractive part of the potential between nanoparticle and monomers, the cut-off distance is set to rc2 = R0 + 2 · 21/6 . The radius of the nano-particle is R0 + σ pm /2, with R0 = 2 and σ pm = 1, the energy parameter pm is equal to 1 and the nano-particle mass is M = 25. The system is simulated in a cubic box with periodic boundary conditions and the larger cut-off distance rc2 is used for all interactions. The box contains K = 1572 chains with N = 10 monomers each. The equilibration is done at constant pressure p = 1 and the temperature ranges from 0.45 to 1.06. For this a Nosé-Hoover thermostat with coupling time constant tNH = 2 is employed. The production runs are then carried out at constant volume and temperature. The monomer number density is therefore temperature dependent and varies from ρ = 0.88 to 1.05, corresponding to box sizes of 26.53 to 24.73 . The simulations are using a time step of δt = 0.0035 and a simulation length between tmax = 35 000 and tmax = 10 000 000.
the shear modulus is the autocorrelation function of the offdiagonal stress tensor components: 1 α β r (t)Fij (t), α = β = x, y, z. (4) αβ (t) = − V i,j ij V is the volume of the simulation box, rijα (t) is the αcomponent of the distance vector between monomer i and β j, Fij (t) is the β-component of the force of monomer j to monomer i and the sum runs over all pairs of particles (monomers and nano-particle). to symmetry, the kinetic Due β part of the stress tensor (∝ viα vi ) cannot contribute to the off-diagonal elements. Numerically its contribution, which we neglect, is below 2%. The stress relaxation modulus is then given as24 Gst (t) =
V (t)xy (0), kB T xy
where the average is composed of the ensemble average and an average over the three off-diagonal elements of the stress tensor. The dynamic behavior of short melt chains is generally described by the Rouse model, which yields a completely intramolecular stress tensor that is given by the time correlation function of the chain eigenmodes Xp : R αβ (t) =
N−1 ρkB T Xpα (t)Xpβ (t) 2 , N p=1 Xpx eq
In our previous paper, we showed in detail how to determine the modulus of a polymer melt from an equilibrium MD simulation by following the suggestions in Ref. 19. We will briefly summarize this here. The microscopic basis of
α, β = x, y, z
(6)
with N 1 (n − 1/2)pπ Xp (t) = , r (t)cos N n=1 n N p = 0, . . . , N − 1,
(7)
with the monomer position vectors rn (t) and where the bar denotes an average over all chains in the system. The stress relaxation function for the Rouse model is then computed as18 GR (t) =
N−1 ρkB T Xpx (t)Xpy (t)Xpx (0)Xpy (0) , 2 2 N p=1 Xpx
(8)
where again an averaging over the different off-diagonal components is implied. From the sum of both G(t) = GR (t) + Gst (t) (τN−1 (T ) − t),
(9)
where τ N − 1 (T) is the shortest relaxation time of the Rouse modes, which is temperature dependent, and (x) is the Heavyside step function, the complex modulus, G*(ω) = G (ω) + iG (ω) (with the storage modulus, G (ω), and the loss modulus, G (ω)), is determined by Fourier transform. For this, the two shear moduli are fitted using the following equations: Gst (t) = Ae−t/τA cos( t) + Be−t/τB ,
III. THE SHEAR MODULUS OF A POLYMER MELT 18
(5)
GR (t) =
4
ai e−bi t .
(10)
(11)
i=1
The first one was suggested by Vladkov and Barrat19 and gives similar fitting qualities for our system (cf. Table I). The
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TABLE I. Fitting parameters for Eq. (10). τA
B
τB
67.2 65.1 67.5 48.6 37.2
0.083 0.084 0.074 0.106 0.179
44.6 43.0 42.8 44.0 43.0
49.5 29.9 16.8 23.3 34.4
8.5 2.5 1.4 0.2 0.1
0.6 0.5 R
0.45 0.50 0.60 0.80 1.06
A
T=0.45 T=0.50 T=0.60 T=0.80 T=1.06
0.7
G (t)
T
0.8
0.4 0.3 0.2 0.1 0
second equation uses only four out of maximal nine modes, so that the number of fitting parameters is manageable. It was checked that this is enough to give reliable fits to the stress relaxation function over the complete time range and temperature range. Figure 1 shows the time dependent shear modulus from direct calculation of the stress tensor components. A strong increase in the relaxation time τ B with decreasing temperature is observed, which is typical for glass-forming fluids. The frequency of the short time oscillations is basically unchanged over the studied temperature range, because they represent the bond length fluctuations of the polymer chains, which are fixed by the given interaction potentials. The increasing relaxation time is also seen in the moduli computed from the Rouse modes (Figure 2). But the order of the zero-time amplitude is reversed: Gst (0) increases and GR (0) decreases with decreasing temperature. So the effect of the short time correction is more pronounced for colder melts, where the “short time” is extended. This is already included in the analysis due to the temperature dependent shortest relaxation time τ N − 1 (T) in Eq. (9). The relaxation times can be found in Table II. Since the time dependent relaxation modulus is connected to the zero-shear viscosity by an integration, ∞ G(t)dt , (12) ηST =
-0.1 10-2
10-1
100
101 t
102
103
FIG. 2. Stress relaxation modulus using Rouse modes GR (t) for the indicated temperatures (logarithmic time axis). The sampled data points have an increasing time lag of 0.0035 for small times to 3.5 for long times and are therefore shown as continuous lines.
a concomitant need for collective motion of the monomers, which protracts all relaxation processes. Figure 3 shows the corresponding storage (a) and loss moduli (b). With decreasing temperature the storage modulus increases and two plateaus emerge. The high-frequency plateau corresponds to the trapping of the monomers by their neighbors, so that they are nearly at a fixed position similar to the atoms in a crystal (glassy region). The low-frequency plateau indicates the increased long-time (or α-) relaxation that is similar to the rubbery region in entangled polymer melts. The maximum in the loss modulus, that shifts to lower frequencies at lower temperatures, shows the transition to the glassy region at high frequencies. The minimum at the highest shown frequencies is due to the resonances of the bond length fluctuations at ω ≈ 50 that result in a maximum of the loss modulus (not shown). In macroscopic shear experiments one would not see those resonances and the minimum would be a transition to another plateau.
0
the evolving plateau in GR (t) leads to a strong increase of the viscosity, when approaching the glass-transition (cf. Table II). The reason for the slowing down of the relaxation modulus between the short time local and the long time structural relaxation is the caging of the monomers by their neighbors and
120
T=0.45 T=0.50 T=0.60 T=0.80 T=1.06
100
60
st
G (t)
80
40 20 0 -20 0
0.2
0.4
0.6
0.8
1
t FIG. 1. Stress tensor based modulus Gst (t) for the indicated temperatures. The sampled data points have a time lag of 0.0035 and are therefore close enough to be shown as continuous lines.
IV. TEMPERATURE DEPENDENT ONE-PARTICLE MICRO-RHEOLOGY
Micro-rheology uses the thermal motion of dispersed particles to study the viscoelastic properties of the medium. This motion is represented by the mean squared displacement (MSD) r(t)2 . Storage and loss modulus can be derived starting from the generalized Stokes-Einstein relation in Laplace space, denoted by a tilde above dependent variables in Eq. (13), 2dkB T s ˜ − Ms , (13) G(s) = νπ R s 2 ˜r (s)2 with the particle-monomer cross section R = R0 + 1, which is the distance where hydrodynamic boundary conditions are applied.18, 25 d is the spatial dimension, M is the particle mass, s the Laplace frequency, and ν is a parameter that depends on the particle’s boundary condition; for stick boundary condition ν = 6 and for slip boundary condition ν = 4. For the derivation of Eq. (13) a generalized Langevin equation to describe the motion of the nano-particle and the simple form of the Stokes friction ζ = νπ ηR is used. With this one includes particle inertia but neglects medium inertia. A derivation
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A. Kuhnhold and W. Paul
J. Chem. Phys. 141, 124907 (2014)
TABLE II. Monomer-density, ρ, diffusion constant, D, zero-shear viscosity from micro-rheological analysis, ηMR , and from stress tensor autocorrelation, ηST a , shortest Rouse modes relaxation time, τ N − 1 , and α-relaxation time, τ α , at the studied temperatures.
T 0.45 0.50 0.60 0.80 1.06 a
D
ρ 1.05 1.03 1.00 0.95 0.88
× 10−7
9.2 1.9 × 10−5 2.0 × 10−4 1.0 × 10−3 3.1 × 10−3
ηMR
ηST
τN − 1
Smon (q = 7, t)
τα SNP (q = 7, t)
SNP (q = 1, t)
11 723 670 78 21 9
3370 391 78 18 9
5009 262 33 7.3 2.8
3500 63 4.2 1.1 0.52
27 755 660 55 9.8 3.6
... 44 867 4704 1316 374
For the integration in Eq. (12) the fitting functions, Eqs. (10) and (11), were used and the upper limit was set to 100 000.
of equations including also medium inertia and its impact on our results can be found in our previous paper,18 where we found that this does not improve our results, and where we concluded that using the simple form is sufficient for our system. The moduli are needed in Fourier instead of Laplace space. For this we use the normalized instantaneous diffusion 2 − D . Defining coefficient18 E(t) = D(t) − D = 2d1 dr(t) dt the Fourier transformation of E(t) as E*(ω) = E (ω) + iE (ω) one gets the storage and loss modulus from Eq. (13): kB T E (ω) ω2 + M , G (ω) = νπ R (D − ωE (ω))2 + (ωE (ω))2 (14) kB T (D − ωE (ω)) ω . (15) G (ω) = νπ R (D − ωE (ω))2 + (ωE (ω))2 102
(a)
G’(ω)
100
10
T=0.45 T=0.50 T=0.60 T=0.80 T=1.06
-2
10-4 10-4
10-3
10-2
10-1
100
101
ω 2
10
G’’(ω)
(b)
100
10
-2
10-4
10-3
10-2
10-1
100
101
ω FIG. 3. Complex modulus from stress relaxation modulus of a polymer melt with an embedded particle at different temperatures. (a) Storage modulus and (b) loss modulus.
A. Mean squared displacement and instantaneous diffusion
The motion of the included nano-particle as described by the mean squared displacement and the instantaneous diffusion coefficient is shown in Fig. 4. At the highest temperatures the mean squared displacement shows a ballistic, an intermediate subdiffusive, and a long time diffusive regime. By lowering the temperature the subdiffusive regime extends and the exponent decreases until a plateau is reached close to the glass transition temperature. The emergence of the plateau is due to the so called caging effect that indicates that the particle is trapped by a cage of monomers around it. This effect is of course also seen in the pure melt, where it has been extensively studied for this model.15 Although the nano-particle can break the cage for long times, the diffusion constant is 4 orders of magnitudes lower than at the highest shown temperature. We find oscillations in the diffusion coefficient at the two lowest temperatures at times between 2 and 20 that correspond to the plateau region and show that this is not ideal, but has some dips. The temperature dependence of the diffusion constant is shown in Fig. 5. It follows the function D(T) = a(T − T*)γ with a = 0.008 ± 0.002, T* = 0.44 ± 0.01, and γ = 2.1 ± 0.2. The critical temperature is in good agreement to the result from mode-coupling-theory (MCT) for the pure melt15 and studies of a similar system with a dumbbell as the probe particle.26 The values of D can be found in Table II.
B. Storage and loss moduli
The storage and loss moduli are calculated using Eqs. (14) and (15) and are shown in Fig. 6. A plateau evolves in the storage modulus for decreasing temperatures showing again the caging effect by the cold polymer chains. In contrast to the melt modulus shown in Fig. 3, there is only one plateau before the terminal relaxation. In a similar frequency region a minimum occurs in the loss modulus, that gets deeper for lower temperatures in accordance with that of the melt modulus, but it occurs at lower frequencies compared to that. A closer look at the MSD reveals that at the lowest temperature the nano-particle moves too little in the accessible time range (r(t)2 < R0 = 2), so it is not probing the global melt moduli, but measures local properties. The micro-rheological measurements do show features connected with the temperature dependent change of rheological properties of the melt, but for low temperatures they do not reveal the true moduli.
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124907-5
A. Kuhnhold and W. Paul
J. Chem. Phys. 141, 124907 (2014)
102
1
G’(ω)
100
(a)
2
t
T=1.06 T=0.80 T=0.60 T=0.50 T=0.45
102
10-2 10
100 T=0.45 T=0.50 T=0.60 T=0.80 T=1.06
t2
-4
10
(a)
10-2
10-1
100
101
102
103
-2
104
10-4
t
10-3
10-2
10-1
100
101
100
101
ω 2
10
(b)
0.008
G’’(ω)
D(t)
(b)
0.004
100
0.000 10-2 10-1
100
101
102
103
104
10-2 10-4
t
10-3
10-2
10-2
FIG. 6. Complex modulus from micro-rheology with an embedded particle at different temperatures. (a) Storage modulus and (b) loss modulus.
D(t)
10-3 10-4
A comparison of the zero-shear viscosities determined from micro-rheological analysis by using the slope of the loss modulus:
10-5 10
-6
G (ω) , ω→0 ω
ηMR = lim
(c) 10
-7
10
-2
10
-1
10
0
1
2
10
10
10
3
10
4
t FIG. 4. Motion of the embedded particle with size R0 = 2, monomer-particle interaction pm = 1 in a polymer melt with N = 10 at different temperatures. (a): Mean squared displacement; (b): instantaneous diffusion coefficient; (c): instantaneous diffusion coefficient in logarithmic scale. We show the frequency of data sampling of the MSD for one example (T = 1.06). We use regions with different but constant time steps to have enough points at short times, but save memory at long times. For the D(t) curves, we interpolated the MSD by spline functions and smoothed the result by running averages. Therefore, we do not show points anymore.
10-2 10-3 D(T)
10-1 ω
10-4 10
-5
10
-6
10-7
D(T)=a (T-T*)γ
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
T FIG. 5. Temperature dependence of the self-diffusion constant of the embedded nano-particle.
(16)
and stress tensor calculation (Eq. (12)) shows that they agree down to T = 0.6 but deviate strongly for the two lower temperatures (Table II). This quantity describes the low frequency limit only but as can be seen in Fig. 7 the high frequency region is very different for both moduli at low temperatures, which is not the case at high temperatures, where we find a perfect agreement over a large frequency range. At first glance, this is astonishing, considering limiting frequencies for the applicability of the microrheological method, which can be derived from the frequency dependent penetration depth of the shear waves generated by the particle motion.7, 8 A high frequency cutoff can be defined by the frequency at which this penetration depth is equal to the particle size (ω = 1.1 for T = 1.06), a low frequency cutoff can be defined by the frequency where the penetration depth is equal to half the system size (ω = 0.1 for T = 1.06). The agreement in Fig. 7 (top) extends beyond both of these values (for the low frequencies similar behavior was found in Ref. 8). We hypothesize that the good agreement in the low frequency regime has the same origin as the ability to reproduce the melt modulus in this regime by using the Rouse model. This model only includes intramolecular momentum transport, limited by the size, Rg , of the molecule. The hydrodynamic calculation of the penetration depth, on the other hand, is based on momentum transport through space of a structureless fluid which
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A. Kuhnhold and W. Paul
J. Chem. Phys. 141, 124907 (2014)
102
0.28 0.24 0.2
0
10
10
T=0.45 T=0.50 T=0.60 T=0.80 T=1.06
s(t)
G’(ω), G’’(ω)
(a)
-2
0.16 0.12 0.08 0.04 0
10-4 10-4
10-3
10-2
10-1
100
101
ω 2
G’(ω), G’’(ω)
(b)
100
-2
10-4
10-1
100
101 t
102
103
FIG. 8. Correlated motion between nearby monomers on different chains (dashed lines) and between the nano-particle and a shell of monomers surrounding it (solid lines) for the studied temperatures.
10
10
10-2
10-3
10-2
10-1
100
101
ω FIG. 7. Storage modulus (solid lines) and loss modulus (dashed lines) from micro-rheology (green, red) and from stress tensor calculation (black) at T = 1.06 (a) and T = 0.6 (b).
is limited by the system size, and it may thus only give a conservative upper bound for the lower limiting frequency. C. Collective intermolecular motion
To get an insight in the temperature dependence of local motions, which are obviously very important for our study, we follow the idea of Morhenn et al., who describe the intermediate relaxation of short chain polymer melts by means of collective intermolecular motions.27 For this the quantity rk (t) · rj (t) smon (t) = (17) |rk (t)||rj (t)| is introduced, which measures correlations in the displacements of monomers of different chains (rk (t) and rj (t) are the displacements of monomers k and j, respectively). The average is taken over all pairs of monomers k and j on distinct chains which are closest to each other at the starting time, and a time average is applied in addition. We introduce a similar quantity for the correlated motion between the nano-particle and the monomers in its vicinity: rNP (t) · rj (t) . (18) sNP (t) = |rNP (t)||rj (t)| Here, rNP (t) is the displacement of the nano-particle and, besides the time average, one averages over all monomers j that are closer to the nano-particle center than R0 + 2. The temperature dependent correlated motion of monomers and nano-particle and monomers is shown in Fig. 8 as dashed
and solid lines, respectively. The degree of correlation in the motion of monomers increases and the correlation decays over longer time ranges with decreasing temperature. For the monomer pairs, the maximum occurs around t = 2 for all temperatures. The build-up of the plateau at low temperatures is directly connected to the caging effect, where the monomers are trapped by their neighbors and hence move in a collective manner over a long time range. At very short times s(t) is almost zero, as expected because of the nearly ballistic motion of each monomer. Except for the highest temperature, the correlation is negative up to around t = 0.1, i.e., the monomers on different chains move on average in opposite directions, when they meet each other first time. However, this effect is very weak and decreases with increasing temperature. For the correlated motion between the nano-particle and the monomers surrounding it, there is no negative short time region at all. The amplitude of sNP (t) is much smaller than that of smon (t); but this decrease can only be analyzed qualitatively, since for smon (t) only two near particles are correlated, whereas for sNP (t) a shell of around 160 monomers around the nano-particle is used. For the highest temperature, the maximum of the nano-particle-monomer correlation occurs more than one decade later than that of the monomer-monomer correlation, which is attributed to the small displacement of the nano-particle that has moved only about the diameter of a monomer at this time. By lowering the temperature, the maximum shifts to shorter times, so that it occurs at the same time as the monomer correlation maximum for T = 0.45. The similarity of the curves means that we have a strong and long lasting collective motion between all close-by particles, i.e., monomers as well as the nano-particle, which means that the vicinity of the nano-particle is not changed over a long period. To obtain an idea of the size of the region of correlated motion we study the distance dependence of the correlation between monomers and the nano-particle at a given time lag τ : rNP (τ ) · rj (τ ) δ(r − |rNP (τ ) − rj (τ )|) . q(r, τ ) = |rNP (τ )||rj (τ )| (19) This is shown in Figure 9 for τ = 35. Independent of temperature, the correlations persist over 7 monomer diameters from the nano-particle surface. For the three lowest
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124907-7
A. Kuhnhold and W. Paul
J. Chem. Phys. 141, 124907 (2014)
0.16
1
T=0.45
0.14
T=0.50
0.12
T=0.60
0.1
T=1.06
S(q,t)
q(r)
0.8
T=0.80
0.08
0.6
0.06
0.4 T=0.45
0.04
0.2 T=0.60
T=0.50
0.02 0
0 2
3
4
5
6
7
8
9
10-3 10-2 10-1 100 101 102 103 104 105 t
10
r
temperatures a layering occurs, that is equivalent to the layering of the monomer density, and that is most pronounced at T = 0.5. This is another indication that the nano-particle sees a long-lived shell of monomers, whose dynamic behavior can be assumed to be different from that of bulk monomers, for temperatures up to 0.6. This leads to the conclusion that over a wide frequency range the micro-rheological moduli will describe local properties and not the bulk behavior, and therefore the moduli in Fig. 7 disagree for frequencies beyond the flow regime. D. Incoherent intermediate scattering function
An experimentally accessible way to study the dynamics of the nano-particle is to look at the incoherent intermediate scattering function, which is defined as (20)
where r(t) is the position of the particle and q is a scattering vector. The brackets denote an ensemble average (calculated as average over starting times). For the monomeric scattering function an additional sum over all NK monomers is required, NK 1 −iq·(rj (t)−rj (0)) e . (21) Smon (q, t) = N K j =1 For the evaluation of these equations q is chosen to be parallel to the x-direction, so that q · (r(t) − r(0)) = q(rx (t) − rx (0)). An advantage of studying this function is its direct accessibility in scattering experiments, with which a qualitative comparison can be made (see below). Figure 10 shows the incoherent intermediate scattering functions of the monomers (dashed lines) and the nanoparticle (solid lines) for the studied temperatures at q = 7, which is close to the first maximum of the structure factor of the polymer model.15 At low temperatures both functions show a two-step decay, where the long-time relaxation corresponds to the structural (α-) relaxation and the short time decay is due to motions in the cage. The relaxation times for the monomers are about one order of magnitude smaller than the corresponding times for the nano-particle. The plateau value is also smaller, which means that the fraction of local relax-
FIG. 10. Incoherent intermediate scattering function of the nano-particle (solid lines) and the monomers (dashed lines) at different temperatures and q = 7.
ation of the melt is higher than that of the particle. This is expected, because the big nano-particle has a much smaller displacement on these time scales than the monomer. The αrelaxation time is defined as S(q, τ α ) = 0.3 and the temperature dependence of the nano-particle and monomer results for this time scale is shown in Figure 11. Similar to the diffusion constant it follows the function τ α (T) = a(T − T*)γ , with a = 0.19 ± 0.02, T* = 0.44 ± 0.01, and γ = −2.0 ± 0.1 for the monomers at q = 7, a = 0.6 ± 0.4, T* = 0.43 ± 0.01, and γ = −2.6 ± 0.4 for the nano-particle at q = 7 and a = 150 ± 10, T* = 0.46 ± 0.02, and γ = −1.7 ± 0.2 for the nanoparticle at q = 1. So both quantities, the diffusion constant and the α-relaxation time, yield the same critical temperature, which is in accordance with former analysis of the glass transition of this model.15, 26 In Ref. 26 a similar polymer model was studied with N ranging from 5 to 25, where the probe particle was a dumbbell consisting of two monomers with a slightly bigger mass and radius compared to the monomers of the polymer chains. This system was chosen to be comparable to recent single molecule spectroscopy results, where this dumbbell models a fluorophore, which is either free or attached to the center or the end of a polymer chain. In all cases Vallée et al. found that the dumbbell clearly probes the glass transition of the polymer matrix, and therefore support the application of single molecule spectroscopy. So our findings of
105
τα(T)=a (T-T*)γ
4
10
3
10 τα(T)
FIG. 9. Distance dependent correlated motion between the nano-particle and the monomers at distance r from the nano-particle center at a lag time τ = 35 for the studied temperatures.
SNP (q, t) = e−iq·(r(t)−r(0)) ,
T=0.80 T=1.06
102 1
10
100 10-1
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
T FIG. 11. Temperature dependence of the α-relaxation time of the monomers (blue) for q = 7 and the embedded nano-particle for q = 7 (red) and q = 1 (green).
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A. Kuhnhold and W. Paul
J. Chem. Phys. 141, 124907 (2014)
TABLE III. Fit parameter for the stretched exponential fit (Eq. (22)) of the α-relaxation in the incoherent intermediate scattering function of the embedded particle at q = 7 in the range 0.2 ≤ S(q, t) ≤ 0.9.
0.8
T
0.6
β
τ
0.9 ± 0.1 0.9 ± 0.1 0.9 ± 0.2 1.0 ± 0.2 1.0 ± 0.2
0.76 ± 0.10 0.76 ± 0.10 0.75 ± 0.28 0.79 ± 0.32 1.02 ± 0.40
24 000 ± 9000 600 ± 200 50 ± 15 8±2 2.9 ± 0.6
S(q,t)
0.45 0.50 0.60 0.80 1.06
c
1
0.4 T=0.45 T=0.50
0.2 T=0.60 0
T=0.80 T=1.06
10-3 10-2 10-1 100 101 102 103 104 105 t
the coupling of a nano-particle to the glass transition behavior of a polymer melt agree well with the results in Ref. 26. Another characteristic temperature that lies below the glass transition temperature Tg in contrast to T*, that lies ∼20% above Tg , is T0 , which is defined by the Vogel-FulcherTammann equation τ α = τ 0 exp (E/(T − T0 )). For the nanoparticle at q = 7 one gets T0 = 0.35 ± 0.02 which again is in agreement with the literature value.15 The α-relaxation part of the scattering function is fitted to a stretched exponential, (22)
The resulting fit parameters for fitting the scattering function of the nano-particle for q = 7 in the range 0.2 ≤ S(q, t) ≤ 0.9 are listed in Table III. The stretching exponent β is always smaller than one within fitting uncertainties, which means that the particle motion is subdiffusive in this regime. This is exactly what the theory predicts12–14 and what is found in experiments and simulations on similar systems. So the suspended nano-particle shows again the right qualitative behavior. However, there are also experiments that seem to reveal a different behavior at low temperatures and small scattering vectors, i.e., large length scales.16, 17 Those experiments are done by XPCS of gold nano-particles with R = 2–3 nm in a polystyrene melt with Mw = 2000 g/mol. They measure the intensity time autocorrelation function g2 (q, t), which is connected to S(q, t) by g2 (q, t) = 1 + b
S(q, t)2 , S(q, 0)2
S(q, t) ≈ e−q
r(t)2 /6
(24)
.
Figure 13 shows the directly calculated MSD (solid lines) together with the MSD calculated from Eq. (24) (dotted lines, q = 7). Here one sees the influence of the numerical limit: To reach a MSD of 5 (4), the scattering function must have a value lower than 2 × 10−18 (7 × 10−15 ), which is almost impossible to achieve. This limit depends on the value of q; with smaller q higher displacements can be calculated, because S(q, t) contains the product of them. For smaller displacements the two curves cannot be distinguished, which is expected for the ballistic and the diffusive regime, where (24) is exact, but not for the subdiffusive motion. Assuming that this is valid for all q-values, one can fit the α-relaxation (22) directly to the mean squared displacement, r(t)2 = 6((t/τ )β − ln c)/q 2 .
(23)
where b is an instrumental parameter, which varies between 0 and 1, and the value of q is varied between 0.04 nm−1 and 0.58 nm−1 . For a temperature of 303 K (10% above Tg ) they find stretching exponents β > 1 for all scattering vectors and conclude on a superdiffusive motion at low temperatures. Our temperature range also covers temperatures down to 10% above Tg (Tg ≈ 0.4114 ), but to be better comparable to those experiments, we would need to study smaller q values. The behavior at smaller momentum transfers q is shown in Fig. 12 where S(q = 7, t) (solid lines) is compared to S(q = 1, t) (dashed lines). The α-relaxation time increases by two orders of magnitude and the plateau value shifts to almost 1 for all temperatures. The fitting for T = 0.45 is therefore not reliable anymore. However, we would have to decrease q even further, to have our R0 q values in the range of Rq in Ref. 16, which would require the determination of extremely long relaxation times which are not reachable in a reasonable CPU
2
(25)
With this one can do the fitting for several q with one MSD dataset. For q < 1 the plateau value c is almost 1, so that the
T=1.06 T=0.80 T=0.60 100 T=0.50 T=0.45 102
2
β
time. For this reason, we use a slightly larger temperature of T = 0.6 and look at the relationship between the incoherent intermediate scattering function of the nano-particle and its mean squared displacement, which is independent of q, in the following. To get the mean squared displacement of the nanoparticle from an (experimentally recorded) incoherent intermediate scattering function, the following Gaussian approximation is often used:
Sα (t) = ce−(t/τ ) .
FIG. 12. Incoherent intermediate scattering function of the nano-particle at different temperatures with q = 7 (solid lines) and q = 1 (dashed lines).
10-2 10
-4
10-2
10-1
100
101
102
103
104
t FIG. 13. Directly calculated MSD (solid lines) and MSD calculated from Eq. (24) (dotted lines) for q = 7.
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124907-9
A. Kuhnhold and W. Paul
J. Chem. Phys. 141, 124907 (2014)
TABLE IV. Fit parameter for the stretched exponential fit (Eq. (25)) of the α-relaxation in the mean squared displacement of the embedded particle at different q at T = 0.6 in the range −6 ln (0.9)/q2 ≤ r(t)2 ≤ −6 ln (0.2)/q2 . q 0.60 0.50 0.40 0.30 0.20 0.10
β 1.00 ± 0.02 1.01 ± 0.03 0.97 ± 0.03 0.98 ± 0.03 1.1 ± 0.1 1.1 ± 0.3
τ 13 600 ± 100 19 600 ± 100 30 700 ± 200 55 900 ± 300 108 000 ± 1000 320 000 ± 10 000
additive term in (25) vanishes. To be better comparable to the results in Refs. 16 and 17, we fit the MSD for q < 0.65. Our results for T = 0.6 (33% above T*) are shown in Table IV. We find stretching exponents of about one to slightly above one for the smallest q-values. Again, we would conclude, that our results give no indication of a superdiffusive motion of the nano-particles at temperatures close to Tg . This is clearly so for q ≥ 0.3, but for q ≤ 0.2 we are susceptible to systematic uncertainties in the MSD used to predict the incoherent scattering. In the experiment, other factors may explain the finding of apparent superdiffusive motion, which is clearly incompatible with all known phenomenology of the glass transition and with our findings in this study in particular. We cannot reasonably speculate about these factors, but we think that in view of the disagreement with all other information available from experiment or simulation, the validity of the conclusion about superdiffusive motion on the basis of the XPCS experiments should be critically reexamined. V. CONCLUSIONS
We have presented a Molecular Dynamics simulation study on the temperature dependence of the motion of a nanoparticle suspended in a polymer melt and the applicability of the micro-rheological approach to the determination of the shear moduli of the polymer melt at low temperatures. The model system consists of short (N = 10) bead-spring polymer chains and is well studied in the literature.15, 19, 26 Following the suggestions of Vladkov and Barrat19 to calculate the complex shear modulus from the stress tensor autocorrelation and the Rouse modes, we showed the temperature dependence of the storage and loss moduli of our system. We found the typical properties of extending β- and α-relaxations with decreasing temperature, which means developing plateaus in the storage modulus and corresponding maxima in the loss modulus. The terms α- and β-relaxation correspond to global, structural and local rearrangements, respectively. The latter are related to the so-called caging effect, where a monomer is trapped by its neighbors. After this we studied the temperature dependent thermal motion of the suspended nano-particle, which also showed a caging effect represented by a plateau in the mean squared displacement. The decrease of the diffusion constant revealed the critical temperature of mode-coupling-theory for our model.26 From the MSD we calculated the complex modulus via the micro-rheological approach and found again prop-
erties of extended relaxations, but occurring at different frequencies compared to the melt moduli. A comparison of the zero-shear viscosities from micro-rheology and stress tensor calculation showed that the micro-rheological approach does not work for T < 0.6. This was explained by the fact that the particle does not explore the bulk, but collectively moves with the cage of particles surrounding its original position. This was confirmed by looking at the cooperative motion of the nano-particle and surrounding monomers in time and space, where we found a correlation in time over three decades and a strong layering effect in space for temperatures lower than or equal to 0.6. For such a case the theory of micro-rheology needs to be adapted, maybe via including the caging effect in an explicit way. In the last part we dealt with the incoherent intermediate scattering function, which showed a two-step decay for large enough q for both monomers and the nano-particle. The relaxation around the plateau is described by the β-relaxation and the long-time decay by the α-relaxation. The latter can be fitted with a stretched exponential function. We have fitted this function for various values of the scattering vector q, using the similarity of the incoherent scattering function with the mean squared displacement. The stretching exponent stayed in all cases around 1 in contrast to experimental findings of XPCS measurements,16, 17 which yielded a superdiffusive motion of the suspended particle at long times. Because of the restrictions of time scales which can be covered in numerical simulations, we cannot fully disprove those findings, but we wanted to point out that they need reexamination in view of all results on the behavior of relaxation functions in a glass forming system. 1 T.
G. Mason and D. A. Weitz, Phys. Rev. Lett. 74, 1250 (1995); T. G. Mason, K. Ganesan, J. H. van Zanten, D. Wirtz, and S. C. Kuo, ibid. 79, 3282 (1997). 2 T. G. Mason, Rheol. Acta 39, 371 (2000). 3 T. M. Squires and T. G. Mason, Annu. Rev. Fluid Mech. 42, 413 (2010). 4 B. U. Felderhof, Physica A 166, 492 (1990); J. Chem. Phys. 123, 044902 (2005); 123, 184903 (2005). 5 S. Jeney, B. Luki´ c, J. A. Kraus, T. Franosch, and L. Forró, Phys. Rev. Lett. 100, 240604 (2008); T. Franosch, M. Grimm, M. Belushkhin, F. M. Mor, G. Foffi, L. Forró, and S. Jeney, Nature 478, 85 (2011); M. Grimm, S. Jeney, and T. Franosch, Soft Matter 7, 2076 (2011); M. Grimm, T. Franosch, and S. Jeney, Phys. Rev. E 86, 021912 (2012). 6 A. Córdoba, J. D. Schieber, and T. Indei, Phys. Fluids 24, 073103 (2012). 7 T. Indei, J. D. Schieber, A. Córdoba, and E. Pilyugina, Phys. Rev. E 85, 021504 (2012); T. Indei, J. D. Schieber, and A. Córdoba, ibid. 85, 041504 (2012). 8 M. Karim, S. C. Kohale, T. Indei, J. D. Schieber, and R. Khare, Phys. Rev. E 86, 051501 (2012). 9 A. Erbas, R. Podgornik and R. R. Netz, Eur. Phys. J. E 32, 147 (2010). 10 J. H. Gibbs and E. A. DiMarzio, J. Chem. Phys. 28, 373 (1958). 11 U. Bengtzelius, Phys. Rev. A 34, 5059 (1986). 12 J. Jäckle, Rep. Prog. Phys. 49, 171 (1986). 13 W. Götze and L. Sjögren, J. Phys. C 20, 879 (1987). 14 K. Binder, J. Baschnagel, and W. Paul, Prog. Polym. Sci. 28, 115 (2003). 15 C. Bennemann, W. Paul, K. Binder, and B. Dünweg, Phys. Rev. E 57, 843 (1998); C. Bennemann, J. Baschnagel, and W. Paul, Eur. Phys. J. B 10, 323 (1999); C. Bennemann, W. Paul, J. Baschnagel, and K. Binder, J. Phys.: Condens. Matter 11, 2179 (1999). 16 R. L. Leheny, Curr. Opin. Colloid Interface Sci. 17, 3 (2012). 17 H. Guo, G. Bourret, M. K. Corbierre, S. Rucareanu, R. B. Lennox, K. Laaziri, L. Piche, M. Sutton, J. L. Harden, and R. L. Leheny, Phys. Rev. Lett. 102, 075702 (2009). 18 A. Kuhnhold and W. Paul, Phys. Rev. E 90, 022602 (2014).
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Vladkov and J.-L. Barrat, Macromol. Theory Simul. 15, 252 (2006). Kremer and G. S. Grest, J. Chem. Phys. 92, 5057 (1990). 21 B. Hess, C. Kutzner, D. van der Spoel, and E. Lindahl, J. Chem. Theory Comput. 4, 435 (2008). 22 D. van der Spoel, E. Lindahl, B. Hess, A. R. van Buuren, E. Apol, P. J. Meulenhoff, D. P. Tieleman, A. L. T. M. Sijbers, K. A. Feenstra, R. van Drunen, and H. J. C. Berendsen, Gromacs User Manual version 4.5.4, 2010, see www.gromacs.org. 23 C. Bennemann, J. Baschnagel, W. Paul, and K. Binder, Comput. Theor. Polym. Sci. 9, 217 (1999). 20 K.
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Doi and S. F. Edwards, The Theory of Polymer Dynamics (Oxford University Press, New York, 1996). 25 J. R. Schmidt and J. L. Skinner, J. Chem. Phys. 119, 8062 (2003). 26 R. A. Vallée, M. Van der Auweraer, W. Paul, and K. Binder, Phys. Rev. Lett. 97, 217801 (2006); Europhys. Lett. 79, 46001 (2007); R. A. Vallée, W. Paul, and K. Binder, J. Chem. Phys. 127, 154903 (2007); 132, 034901 (2010); R. A. Vallée, W. Paul, and K. Binder, Macromolecules 43, 10714 (2010). 27 H. Morhenn, S. Busch, H. Meyer, D. Richter, W. Petry, and T. Unruh, Phys. Rev. Lett. 111, 173003 (2013).
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THE JOURNAL OF CHEMICAL PHYSICS 141, 124907 (2014)
Temperature dependent micro-rheology of a glass-forming polymer melt studied by molecular dynamics simulation A. Kuhnholda) and W. Paul Institut für Physik, Martin-Luther-Universität Halle-Wittenberg, 06099 Halle (Saale), Germany
(Received 21 May 2014; accepted 9 September 2014; published online 26 September 2014) We present a Molecular Dynamics simulation study of a micro-rheological probing of the glass transition in a polymer melt. Our model system consists of short bead-spring chains and the temperature ranges from well above the glass transition temperature to about 10% above it. The nano-particle clearly couples to the slowing down of the polymer segments and the calculated storage and loss moduli reveal the approach to the glass transition. At temperatures close to the mode coupling Tc of the polymer melt, the micro-rheological moduli measure the local viscoelastic response of the cage of monomers surrounding the nano-particle and no longer reveal the true melt moduli. The incoherent scattering function of the nano-particle exhibits a stretched exponential decay, typical for the α-process in glass forming systems. We find no indication of a strong superdiffusive regime as has been deduced from a recent experiment in the same temperature range but for smaller momentum transfers. © 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4896151] I. INTRODUCTION
Passive micro-rheology has become the method of choice for studying linear viscoelastic properties of soft matter especially for small and biological samples, because of the possibility to receive information in a non-invasive/non-destructive manner, by studying the Brownian motion of embedded (sub-)micron sized particles. This motion is connected to the complex modulus of the sample in a wide frequency regime. The underlying theories for the analysis of the Brownian motion range from a rather simple Generalized Stokes-Einstein relation1–3 to full hydrodynamic approaches.4–9 When approaching the glass transition temperature, the mobility of polymeric liquids decreases by orders of magnitude. The glass transition itself is still not fully understood and has found much theoretical and experimental interest.10–14 The glass transition of a pure polymeric system of a model similar to that used here has already been studied in great detail.14, 15 The interesting point is now to see how dispersed nano-particles couple to the decreasing mobility of the polymer chains, which is theoretically described by the so-called caging effect. Many experiments in this field are done by optical measurements, e.g., diffusing wave spectroscopy (DWS) and dynamic light scattering (DLS), where the quantity of interest is the incoherent intermediate scattering function of dispersed micron sized particles. Recent studies with suspended gold nano-particles measured by X-ray photon correlation spectroscopy (XPCS) find a compressed exponential behavior of the incoherent intermediate scattering function with exponent larger than 1, which would indicate a superdiffusive motion of the nano-particles at long times.16, 17 By using a simple Gaussian approximation to obtain the mean squared displacement from the scattering function, they also look at rheological properties such as the temperature dependent viscosity, which is a good predictor for the glass transition. a) Electronic mail: [email protected]
0021-9606/2014/141(12)/124907/10/$30.00
In a previous publication,18 we showed for the same model as is studied in this work that the micro-rheological approach does work very well over a wide frequency range if the particle-monomer-interaction is not too strong. We carefully examined the effect of analyzing the particle mean squared displacement on several levels of theoretical approximation, ranging from the original Stokes-Einstein approach suggested by Mason and Weitz,1 to a complete consideration of hydrodynamic effects including fluid inertia and vortex generation by the particle motion. Including the latter two effects proved to be necessary for strong attractive interactions between the nano-particle and the monomers to avoid unphysical behavior of the obtained moduli. For relatively weak attraction the original Stokes-Einstein level description including the inertia term for the nano-particle provided an almost quantitative estimate for the melt moduli. For this reason we chose here a rather weak attraction between nano-particle and polymer melt to study the influence of lowering the temperature on the results of such micro-rheological measurements. We will first introduce the model that we use for our simulation study in Sec. II. After this we will show the determination of the temperature dependent melt modulus following the method introduced in Ref. 19 in Sec. III. Section IV then presents the micro-rheological approach to obtain the melt modulus from the Brownian motion of the suspended nano-particle. To compare our results to recent experiment we also look at the incoherent intermediate scattering function of the nano-particle in this section. Finally, Sec. V will offer our conclusions.
II. MODEL AND SIMULATION
For the simulations, we use a bead-spring polymer model, that is similar to that of Kremer and Grest,20 where all pairs of monomers interact via a truncated and shifted LJ and where the bonds are Lennard-Jones potential Umm
141, 124907-1
© 2014 AIP Publishing LLC
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124907-2
A. Kuhnhold and W. Paul
J. Chem. Phys. 141, 124907 (2014)
included by a FENE potential UF with a spring constant k = 30 and a maximum elongation RF = 1.5σ mm , where σ mm is the diameter of a bead: 12 6 σmm σ 4 + Ush (r), r ≤ rc1 − mm mm LJ r r , Umm (r) = 0, r > rc1 (1)
k 2 r 2 UF (r) = − RF ln 1 − . 2 RF
(2)
To have vanishing potential, force and first derivative of force at the cut-off distance rc1 , the shift function Ush is added. The form and derivation of this function can be found in the manual of the GROMACS-4.5.4 Molecular Dynamics simulation package, which is used for the simulations.21, 22 In the original Kremer-Grest model only the repulsive part of the LennardJones potential is used by setting the cut-off distance to rc1 = 21/6 σ mm . For the investigation of temperature dependent behavior, it is mandatory to use also the attractive part of the potential by setting rc1 = 2 · 21/6 σ mm , so that the model is no longer athermal.23 In the following we use reduced LennardJones units for all numerical values, i.e., mm = 1, σ mm = 1, σ m = 1, τ = √ mm = 1. For studying the micro-rheology of mm /m
our system, we add a nano-particle, which is modeled as a hard sphere of radius R0 , that interacts with the monomers by a Lennard-Jones potential similar to that in Eq. (1), but with a hard sphere shift in the denominator: ⎧ σ 6 σpm 12 ⎨ pm +Ush (r), r ≤ rc2 4pm r−R − r−R LJ (r) = . Upm 0 0 ⎩ 0, r > rc2 (3) To have also the attractive part of the potential between nanoparticle and monomers, the cut-off distance is set to rc2 = R0 + 2 · 21/6 . The radius of the nano-particle is R0 + σ pm /2, with R0 = 2 and σ pm = 1, the energy parameter pm is equal to 1 and the nano-particle mass is M = 25. The system is simulated in a cubic box with periodic boundary conditions and the larger cut-off distance rc2 is used for all interactions. The box contains K = 1572 chains with N = 10 monomers each. The equilibration is done at constant pressure p = 1 and the temperature ranges from 0.45 to 1.06. For this a Nosé-Hoover thermostat with coupling time constant tNH = 2 is employed. The production runs are then carried out at constant volume and temperature. The monomer number density is therefore temperature dependent and varies from ρ = 0.88 to 1.05, corresponding to box sizes of 26.53 to 24.73 . The simulations are using a time step of δt = 0.0035 and a simulation length between tmax = 35 000 and tmax = 10 000 000.
the shear modulus is the autocorrelation function of the offdiagonal stress tensor components: 1 α β r (t)Fij (t), α = β = x, y, z. (4) αβ (t) = − V i,j ij V is the volume of the simulation box, rijα (t) is the αcomponent of the distance vector between monomer i and β j, Fij (t) is the β-component of the force of monomer j to monomer i and the sum runs over all pairs of particles (monomers and nano-particle). to symmetry, the kinetic Due β part of the stress tensor (∝ viα vi ) cannot contribute to the off-diagonal elements. Numerically its contribution, which we neglect, is below 2%. The stress relaxation modulus is then given as24 Gst (t) =
V (t)xy (0), kB T xy
where the average is composed of the ensemble average and an average over the three off-diagonal elements of the stress tensor. The dynamic behavior of short melt chains is generally described by the Rouse model, which yields a completely intramolecular stress tensor that is given by the time correlation function of the chain eigenmodes Xp : R αβ (t) =
N−1 ρkB T Xpα (t)Xpβ (t) 2 , N p=1 Xpx eq
In our previous paper, we showed in detail how to determine the modulus of a polymer melt from an equilibrium MD simulation by following the suggestions in Ref. 19. We will briefly summarize this here. The microscopic basis of
α, β = x, y, z
(6)
with N 1 (n − 1/2)pπ Xp (t) = , r (t)cos N n=1 n N p = 0, . . . , N − 1,
(7)
with the monomer position vectors rn (t) and where the bar denotes an average over all chains in the system. The stress relaxation function for the Rouse model is then computed as18 GR (t) =
N−1 ρkB T Xpx (t)Xpy (t)Xpx (0)Xpy (0) , 2 2 N p=1 Xpx
(8)
where again an averaging over the different off-diagonal components is implied. From the sum of both G(t) = GR (t) + Gst (t) (τN−1 (T ) − t),
(9)
where τ N − 1 (T) is the shortest relaxation time of the Rouse modes, which is temperature dependent, and (x) is the Heavyside step function, the complex modulus, G*(ω) = G (ω) + iG (ω) (with the storage modulus, G (ω), and the loss modulus, G (ω)), is determined by Fourier transform. For this, the two shear moduli are fitted using the following equations: Gst (t) = Ae−t/τA cos( t) + Be−t/τB ,
III. THE SHEAR MODULUS OF A POLYMER MELT 18
(5)
GR (t) =
4
ai e−bi t .
(10)
(11)
i=1
The first one was suggested by Vladkov and Barrat19 and gives similar fitting qualities for our system (cf. Table I). The
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124907-3
A. Kuhnhold and W. Paul
J. Chem. Phys. 141, 124907 (2014)
TABLE I. Fitting parameters for Eq. (10). τA
B
τB
67.2 65.1 67.5 48.6 37.2
0.083 0.084 0.074 0.106 0.179
44.6 43.0 42.8 44.0 43.0
49.5 29.9 16.8 23.3 34.4
8.5 2.5 1.4 0.2 0.1
0.6 0.5 R
0.45 0.50 0.60 0.80 1.06
A
T=0.45 T=0.50 T=0.60 T=0.80 T=1.06
0.7
G (t)
T
0.8
0.4 0.3 0.2 0.1 0
second equation uses only four out of maximal nine modes, so that the number of fitting parameters is manageable. It was checked that this is enough to give reliable fits to the stress relaxation function over the complete time range and temperature range. Figure 1 shows the time dependent shear modulus from direct calculation of the stress tensor components. A strong increase in the relaxation time τ B with decreasing temperature is observed, which is typical for glass-forming fluids. The frequency of the short time oscillations is basically unchanged over the studied temperature range, because they represent the bond length fluctuations of the polymer chains, which are fixed by the given interaction potentials. The increasing relaxation time is also seen in the moduli computed from the Rouse modes (Figure 2). But the order of the zero-time amplitude is reversed: Gst (0) increases and GR (0) decreases with decreasing temperature. So the effect of the short time correction is more pronounced for colder melts, where the “short time” is extended. This is already included in the analysis due to the temperature dependent shortest relaxation time τ N − 1 (T) in Eq. (9). The relaxation times can be found in Table II. Since the time dependent relaxation modulus is connected to the zero-shear viscosity by an integration, ∞ G(t)dt , (12) ηST =
-0.1 10-2
10-1
100
101 t
102
103
FIG. 2. Stress relaxation modulus using Rouse modes GR (t) for the indicated temperatures (logarithmic time axis). The sampled data points have an increasing time lag of 0.0035 for small times to 3.5 for long times and are therefore shown as continuous lines.
a concomitant need for collective motion of the monomers, which protracts all relaxation processes. Figure 3 shows the corresponding storage (a) and loss moduli (b). With decreasing temperature the storage modulus increases and two plateaus emerge. The high-frequency plateau corresponds to the trapping of the monomers by their neighbors, so that they are nearly at a fixed position similar to the atoms in a crystal (glassy region). The low-frequency plateau indicates the increased long-time (or α-) relaxation that is similar to the rubbery region in entangled polymer melts. The maximum in the loss modulus, that shifts to lower frequencies at lower temperatures, shows the transition to the glassy region at high frequencies. The minimum at the highest shown frequencies is due to the resonances of the bond length fluctuations at ω ≈ 50 that result in a maximum of the loss modulus (not shown). In macroscopic shear experiments one would not see those resonances and the minimum would be a transition to another plateau.
0
the evolving plateau in GR (t) leads to a strong increase of the viscosity, when approaching the glass-transition (cf. Table II). The reason for the slowing down of the relaxation modulus between the short time local and the long time structural relaxation is the caging of the monomers by their neighbors and
120
T=0.45 T=0.50 T=0.60 T=0.80 T=1.06
100
60
st
G (t)
80
40 20 0 -20 0
0.2
0.4
0.6
0.8
1
t FIG. 1. Stress tensor based modulus Gst (t) for the indicated temperatures. The sampled data points have a time lag of 0.0035 and are therefore close enough to be shown as continuous lines.
IV. TEMPERATURE DEPENDENT ONE-PARTICLE MICRO-RHEOLOGY
Micro-rheology uses the thermal motion of dispersed particles to study the viscoelastic properties of the medium. This motion is represented by the mean squared displacement (MSD) r(t)2 . Storage and loss modulus can be derived starting from the generalized Stokes-Einstein relation in Laplace space, denoted by a tilde above dependent variables in Eq. (13), 2dkB T s ˜ − Ms , (13) G(s) = νπ R s 2 ˜r (s)2 with the particle-monomer cross section R = R0 + 1, which is the distance where hydrodynamic boundary conditions are applied.18, 25 d is the spatial dimension, M is the particle mass, s the Laplace frequency, and ν is a parameter that depends on the particle’s boundary condition; for stick boundary condition ν = 6 and for slip boundary condition ν = 4. For the derivation of Eq. (13) a generalized Langevin equation to describe the motion of the nano-particle and the simple form of the Stokes friction ζ = νπ ηR is used. With this one includes particle inertia but neglects medium inertia. A derivation
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124907-4
A. Kuhnhold and W. Paul
J. Chem. Phys. 141, 124907 (2014)
TABLE II. Monomer-density, ρ, diffusion constant, D, zero-shear viscosity from micro-rheological analysis, ηMR , and from stress tensor autocorrelation, ηST a , shortest Rouse modes relaxation time, τ N − 1 , and α-relaxation time, τ α , at the studied temperatures.
T 0.45 0.50 0.60 0.80 1.06 a
D
ρ 1.05 1.03 1.00 0.95 0.88
× 10−7
9.2 1.9 × 10−5 2.0 × 10−4 1.0 × 10−3 3.1 × 10−3
ηMR
ηST
τN − 1
Smon (q = 7, t)
τα SNP (q = 7, t)
SNP (q = 1, t)
11 723 670 78 21 9
3370 391 78 18 9
5009 262 33 7.3 2.8
3500 63 4.2 1.1 0.52
27 755 660 55 9.8 3.6
... 44 867 4704 1316 374
For the integration in Eq. (12) the fitting functions, Eqs. (10) and (11), were used and the upper limit was set to 100 000.
of equations including also medium inertia and its impact on our results can be found in our previous paper,18 where we found that this does not improve our results, and where we concluded that using the simple form is sufficient for our system. The moduli are needed in Fourier instead of Laplace space. For this we use the normalized instantaneous diffusion 2 − D . Defining coefficient18 E(t) = D(t) − D = 2d1 dr(t) dt the Fourier transformation of E(t) as E*(ω) = E (ω) + iE (ω) one gets the storage and loss modulus from Eq. (13): kB T E (ω) ω2 + M , G (ω) = νπ R (D − ωE (ω))2 + (ωE (ω))2 (14) kB T (D − ωE (ω)) ω . (15) G (ω) = νπ R (D − ωE (ω))2 + (ωE (ω))2 102
(a)
G’(ω)
100
10
T=0.45 T=0.50 T=0.60 T=0.80 T=1.06
-2
10-4 10-4
10-3
10-2
10-1
100
101
ω 2
10
G’’(ω)
(b)
100
10
-2
10-4
10-3
10-2
10-1
100
101
ω FIG. 3. Complex modulus from stress relaxation modulus of a polymer melt with an embedded particle at different temperatures. (a) Storage modulus and (b) loss modulus.
A. Mean squared displacement and instantaneous diffusion
The motion of the included nano-particle as described by the mean squared displacement and the instantaneous diffusion coefficient is shown in Fig. 4. At the highest temperatures the mean squared displacement shows a ballistic, an intermediate subdiffusive, and a long time diffusive regime. By lowering the temperature the subdiffusive regime extends and the exponent decreases until a plateau is reached close to the glass transition temperature. The emergence of the plateau is due to the so called caging effect that indicates that the particle is trapped by a cage of monomers around it. This effect is of course also seen in the pure melt, where it has been extensively studied for this model.15 Although the nano-particle can break the cage for long times, the diffusion constant is 4 orders of magnitudes lower than at the highest shown temperature. We find oscillations in the diffusion coefficient at the two lowest temperatures at times between 2 and 20 that correspond to the plateau region and show that this is not ideal, but has some dips. The temperature dependence of the diffusion constant is shown in Fig. 5. It follows the function D(T) = a(T − T*)γ with a = 0.008 ± 0.002, T* = 0.44 ± 0.01, and γ = 2.1 ± 0.2. The critical temperature is in good agreement to the result from mode-coupling-theory (MCT) for the pure melt15 and studies of a similar system with a dumbbell as the probe particle.26 The values of D can be found in Table II.
B. Storage and loss moduli
The storage and loss moduli are calculated using Eqs. (14) and (15) and are shown in Fig. 6. A plateau evolves in the storage modulus for decreasing temperatures showing again the caging effect by the cold polymer chains. In contrast to the melt modulus shown in Fig. 3, there is only one plateau before the terminal relaxation. In a similar frequency region a minimum occurs in the loss modulus, that gets deeper for lower temperatures in accordance with that of the melt modulus, but it occurs at lower frequencies compared to that. A closer look at the MSD reveals that at the lowest temperature the nano-particle moves too little in the accessible time range (r(t)2 < R0 = 2), so it is not probing the global melt moduli, but measures local properties. The micro-rheological measurements do show features connected with the temperature dependent change of rheological properties of the melt, but for low temperatures they do not reveal the true moduli.
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124907-5
A. Kuhnhold and W. Paul
J. Chem. Phys. 141, 124907 (2014)
102
1
G’(ω)
100
(a)
2
t
T=1.06 T=0.80 T=0.60 T=0.50 T=0.45
102
10-2 10
100 T=0.45 T=0.50 T=0.60 T=0.80 T=1.06
t2
-4
10
(a)
10-2
10-1
100
101
102
103
-2
104
10-4
t
10-3
10-2
10-1
100
101
100
101
ω 2
10
(b)
0.008
G’’(ω)
D(t)
(b)
0.004
100
0.000 10-2 10-1
100
101
102
103
104
10-2 10-4
t
10-3
10-2
10-2
FIG. 6. Complex modulus from micro-rheology with an embedded particle at different temperatures. (a) Storage modulus and (b) loss modulus.
D(t)
10-3 10-4
A comparison of the zero-shear viscosities determined from micro-rheological analysis by using the slope of the loss modulus:
10-5 10
-6
G (ω) , ω→0 ω
ηMR = lim
(c) 10
-7
10
-2
10
-1
10
0
1
2
10
10
10
3
10
4
t FIG. 4. Motion of the embedded particle with size R0 = 2, monomer-particle interaction pm = 1 in a polymer melt with N = 10 at different temperatures. (a): Mean squared displacement; (b): instantaneous diffusion coefficient; (c): instantaneous diffusion coefficient in logarithmic scale. We show the frequency of data sampling of the MSD for one example (T = 1.06). We use regions with different but constant time steps to have enough points at short times, but save memory at long times. For the D(t) curves, we interpolated the MSD by spline functions and smoothed the result by running averages. Therefore, we do not show points anymore.
10-2 10-3 D(T)
10-1 ω
10-4 10
-5
10
-6
10-7
D(T)=a (T-T*)γ
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
T FIG. 5. Temperature dependence of the self-diffusion constant of the embedded nano-particle.
(16)
and stress tensor calculation (Eq. (12)) shows that they agree down to T = 0.6 but deviate strongly for the two lower temperatures (Table II). This quantity describes the low frequency limit only but as can be seen in Fig. 7 the high frequency region is very different for both moduli at low temperatures, which is not the case at high temperatures, where we find a perfect agreement over a large frequency range. At first glance, this is astonishing, considering limiting frequencies for the applicability of the microrheological method, which can be derived from the frequency dependent penetration depth of the shear waves generated by the particle motion.7, 8 A high frequency cutoff can be defined by the frequency at which this penetration depth is equal to the particle size (ω = 1.1 for T = 1.06), a low frequency cutoff can be defined by the frequency where the penetration depth is equal to half the system size (ω = 0.1 for T = 1.06). The agreement in Fig. 7 (top) extends beyond both of these values (for the low frequencies similar behavior was found in Ref. 8). We hypothesize that the good agreement in the low frequency regime has the same origin as the ability to reproduce the melt modulus in this regime by using the Rouse model. This model only includes intramolecular momentum transport, limited by the size, Rg , of the molecule. The hydrodynamic calculation of the penetration depth, on the other hand, is based on momentum transport through space of a structureless fluid which
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124907-6
A. Kuhnhold and W. Paul
J. Chem. Phys. 141, 124907 (2014)
102
0.28 0.24 0.2
0
10
10
T=0.45 T=0.50 T=0.60 T=0.80 T=1.06
s(t)
G’(ω), G’’(ω)
(a)
-2
0.16 0.12 0.08 0.04 0
10-4 10-4
10-3
10-2
10-1
100
101
ω 2
G’(ω), G’’(ω)
(b)
100
-2
10-4
10-1
100
101 t
102
103
FIG. 8. Correlated motion between nearby monomers on different chains (dashed lines) and between the nano-particle and a shell of monomers surrounding it (solid lines) for the studied temperatures.
10
10
10-2
10-3
10-2
10-1
100
101
ω FIG. 7. Storage modulus (solid lines) and loss modulus (dashed lines) from micro-rheology (green, red) and from stress tensor calculation (black) at T = 1.06 (a) and T = 0.6 (b).
is limited by the system size, and it may thus only give a conservative upper bound for the lower limiting frequency. C. Collective intermolecular motion
To get an insight in the temperature dependence of local motions, which are obviously very important for our study, we follow the idea of Morhenn et al., who describe the intermediate relaxation of short chain polymer melts by means of collective intermolecular motions.27 For this the quantity rk (t) · rj (t) smon (t) = (17) |rk (t)||rj (t)| is introduced, which measures correlations in the displacements of monomers of different chains (rk (t) and rj (t) are the displacements of monomers k and j, respectively). The average is taken over all pairs of monomers k and j on distinct chains which are closest to each other at the starting time, and a time average is applied in addition. We introduce a similar quantity for the correlated motion between the nano-particle and the monomers in its vicinity: rNP (t) · rj (t) . (18) sNP (t) = |rNP (t)||rj (t)| Here, rNP (t) is the displacement of the nano-particle and, besides the time average, one averages over all monomers j that are closer to the nano-particle center than R0 + 2. The temperature dependent correlated motion of monomers and nano-particle and monomers is shown in Fig. 8 as dashed
and solid lines, respectively. The degree of correlation in the motion of monomers increases and the correlation decays over longer time ranges with decreasing temperature. For the monomer pairs, the maximum occurs around t = 2 for all temperatures. The build-up of the plateau at low temperatures is directly connected to the caging effect, where the monomers are trapped by their neighbors and hence move in a collective manner over a long time range. At very short times s(t) is almost zero, as expected because of the nearly ballistic motion of each monomer. Except for the highest temperature, the correlation is negative up to around t = 0.1, i.e., the monomers on different chains move on average in opposite directions, when they meet each other first time. However, this effect is very weak and decreases with increasing temperature. For the correlated motion between the nano-particle and the monomers surrounding it, there is no negative short time region at all. The amplitude of sNP (t) is much smaller than that of smon (t); but this decrease can only be analyzed qualitatively, since for smon (t) only two near particles are correlated, whereas for sNP (t) a shell of around 160 monomers around the nano-particle is used. For the highest temperature, the maximum of the nano-particle-monomer correlation occurs more than one decade later than that of the monomer-monomer correlation, which is attributed to the small displacement of the nano-particle that has moved only about the diameter of a monomer at this time. By lowering the temperature, the maximum shifts to shorter times, so that it occurs at the same time as the monomer correlation maximum for T = 0.45. The similarity of the curves means that we have a strong and long lasting collective motion between all close-by particles, i.e., monomers as well as the nano-particle, which means that the vicinity of the nano-particle is not changed over a long period. To obtain an idea of the size of the region of correlated motion we study the distance dependence of the correlation between monomers and the nano-particle at a given time lag τ : rNP (τ ) · rj (τ ) δ(r − |rNP (τ ) − rj (τ )|) . q(r, τ ) = |rNP (τ )||rj (τ )| (19) This is shown in Figure 9 for τ = 35. Independent of temperature, the correlations persist over 7 monomer diameters from the nano-particle surface. For the three lowest
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124907-7
A. Kuhnhold and W. Paul
J. Chem. Phys. 141, 124907 (2014)
0.16
1
T=0.45
0.14
T=0.50
0.12
T=0.60
0.1
T=1.06
S(q,t)
q(r)
0.8
T=0.80
0.08
0.6
0.06
0.4 T=0.45
0.04
0.2 T=0.60
T=0.50
0.02 0
0 2
3
4
5
6
7
8
9
10-3 10-2 10-1 100 101 102 103 104 105 t
10
r
temperatures a layering occurs, that is equivalent to the layering of the monomer density, and that is most pronounced at T = 0.5. This is another indication that the nano-particle sees a long-lived shell of monomers, whose dynamic behavior can be assumed to be different from that of bulk monomers, for temperatures up to 0.6. This leads to the conclusion that over a wide frequency range the micro-rheological moduli will describe local properties and not the bulk behavior, and therefore the moduli in Fig. 7 disagree for frequencies beyond the flow regime. D. Incoherent intermediate scattering function
An experimentally accessible way to study the dynamics of the nano-particle is to look at the incoherent intermediate scattering function, which is defined as (20)
where r(t) is the position of the particle and q is a scattering vector. The brackets denote an ensemble average (calculated as average over starting times). For the monomeric scattering function an additional sum over all NK monomers is required, NK 1 −iq·(rj (t)−rj (0)) e . (21) Smon (q, t) = N K j =1 For the evaluation of these equations q is chosen to be parallel to the x-direction, so that q · (r(t) − r(0)) = q(rx (t) − rx (0)). An advantage of studying this function is its direct accessibility in scattering experiments, with which a qualitative comparison can be made (see below). Figure 10 shows the incoherent intermediate scattering functions of the monomers (dashed lines) and the nanoparticle (solid lines) for the studied temperatures at q = 7, which is close to the first maximum of the structure factor of the polymer model.15 At low temperatures both functions show a two-step decay, where the long-time relaxation corresponds to the structural (α-) relaxation and the short time decay is due to motions in the cage. The relaxation times for the monomers are about one order of magnitude smaller than the corresponding times for the nano-particle. The plateau value is also smaller, which means that the fraction of local relax-
FIG. 10. Incoherent intermediate scattering function of the nano-particle (solid lines) and the monomers (dashed lines) at different temperatures and q = 7.
ation of the melt is higher than that of the particle. This is expected, because the big nano-particle has a much smaller displacement on these time scales than the monomer. The αrelaxation time is defined as S(q, τ α ) = 0.3 and the temperature dependence of the nano-particle and monomer results for this time scale is shown in Figure 11. Similar to the diffusion constant it follows the function τ α (T) = a(T − T*)γ , with a = 0.19 ± 0.02, T* = 0.44 ± 0.01, and γ = −2.0 ± 0.1 for the monomers at q = 7, a = 0.6 ± 0.4, T* = 0.43 ± 0.01, and γ = −2.6 ± 0.4 for the nano-particle at q = 7 and a = 150 ± 10, T* = 0.46 ± 0.02, and γ = −1.7 ± 0.2 for the nanoparticle at q = 1. So both quantities, the diffusion constant and the α-relaxation time, yield the same critical temperature, which is in accordance with former analysis of the glass transition of this model.15, 26 In Ref. 26 a similar polymer model was studied with N ranging from 5 to 25, where the probe particle was a dumbbell consisting of two monomers with a slightly bigger mass and radius compared to the monomers of the polymer chains. This system was chosen to be comparable to recent single molecule spectroscopy results, where this dumbbell models a fluorophore, which is either free or attached to the center or the end of a polymer chain. In all cases Vallée et al. found that the dumbbell clearly probes the glass transition of the polymer matrix, and therefore support the application of single molecule spectroscopy. So our findings of
105
τα(T)=a (T-T*)γ
4
10
3
10 τα(T)
FIG. 9. Distance dependent correlated motion between the nano-particle and the monomers at distance r from the nano-particle center at a lag time τ = 35 for the studied temperatures.
SNP (q, t) = e−iq·(r(t)−r(0)) ,
T=0.80 T=1.06
102 1
10
100 10-1
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
T FIG. 11. Temperature dependence of the α-relaxation time of the monomers (blue) for q = 7 and the embedded nano-particle for q = 7 (red) and q = 1 (green).
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124907-8
A. Kuhnhold and W. Paul
J. Chem. Phys. 141, 124907 (2014)
TABLE III. Fit parameter for the stretched exponential fit (Eq. (22)) of the α-relaxation in the incoherent intermediate scattering function of the embedded particle at q = 7 in the range 0.2 ≤ S(q, t) ≤ 0.9.
0.8
T
0.6
β
τ
0.9 ± 0.1 0.9 ± 0.1 0.9 ± 0.2 1.0 ± 0.2 1.0 ± 0.2
0.76 ± 0.10 0.76 ± 0.10 0.75 ± 0.28 0.79 ± 0.32 1.02 ± 0.40
24 000 ± 9000 600 ± 200 50 ± 15 8±2 2.9 ± 0.6
S(q,t)
0.45 0.50 0.60 0.80 1.06
c
1
0.4 T=0.45 T=0.50
0.2 T=0.60 0
T=0.80 T=1.06
10-3 10-2 10-1 100 101 102 103 104 105 t
the coupling of a nano-particle to the glass transition behavior of a polymer melt agree well with the results in Ref. 26. Another characteristic temperature that lies below the glass transition temperature Tg in contrast to T*, that lies ∼20% above Tg , is T0 , which is defined by the Vogel-FulcherTammann equation τ α = τ 0 exp (E/(T − T0 )). For the nanoparticle at q = 7 one gets T0 = 0.35 ± 0.02 which again is in agreement with the literature value.15 The α-relaxation part of the scattering function is fitted to a stretched exponential, (22)
The resulting fit parameters for fitting the scattering function of the nano-particle for q = 7 in the range 0.2 ≤ S(q, t) ≤ 0.9 are listed in Table III. The stretching exponent β is always smaller than one within fitting uncertainties, which means that the particle motion is subdiffusive in this regime. This is exactly what the theory predicts12–14 and what is found in experiments and simulations on similar systems. So the suspended nano-particle shows again the right qualitative behavior. However, there are also experiments that seem to reveal a different behavior at low temperatures and small scattering vectors, i.e., large length scales.16, 17 Those experiments are done by XPCS of gold nano-particles with R = 2–3 nm in a polystyrene melt with Mw = 2000 g/mol. They measure the intensity time autocorrelation function g2 (q, t), which is connected to S(q, t) by g2 (q, t) = 1 + b
S(q, t)2 , S(q, 0)2
S(q, t) ≈ e−q
r(t)2 /6
(24)
.
Figure 13 shows the directly calculated MSD (solid lines) together with the MSD calculated from Eq. (24) (dotted lines, q = 7). Here one sees the influence of the numerical limit: To reach a MSD of 5 (4), the scattering function must have a value lower than 2 × 10−18 (7 × 10−15 ), which is almost impossible to achieve. This limit depends on the value of q; with smaller q higher displacements can be calculated, because S(q, t) contains the product of them. For smaller displacements the two curves cannot be distinguished, which is expected for the ballistic and the diffusive regime, where (24) is exact, but not for the subdiffusive motion. Assuming that this is valid for all q-values, one can fit the α-relaxation (22) directly to the mean squared displacement, r(t)2 = 6((t/τ )β − ln c)/q 2 .
(23)
where b is an instrumental parameter, which varies between 0 and 1, and the value of q is varied between 0.04 nm−1 and 0.58 nm−1 . For a temperature of 303 K (10% above Tg ) they find stretching exponents β > 1 for all scattering vectors and conclude on a superdiffusive motion at low temperatures. Our temperature range also covers temperatures down to 10% above Tg (Tg ≈ 0.4114 ), but to be better comparable to those experiments, we would need to study smaller q values. The behavior at smaller momentum transfers q is shown in Fig. 12 where S(q = 7, t) (solid lines) is compared to S(q = 1, t) (dashed lines). The α-relaxation time increases by two orders of magnitude and the plateau value shifts to almost 1 for all temperatures. The fitting for T = 0.45 is therefore not reliable anymore. However, we would have to decrease q even further, to have our R0 q values in the range of Rq in Ref. 16, which would require the determination of extremely long relaxation times which are not reachable in a reasonable CPU
2
(25)
With this one can do the fitting for several q with one MSD dataset. For q < 1 the plateau value c is almost 1, so that the
T=1.06 T=0.80 T=0.60 100 T=0.50 T=0.45 102
2
β
time. For this reason, we use a slightly larger temperature of T = 0.6 and look at the relationship between the incoherent intermediate scattering function of the nano-particle and its mean squared displacement, which is independent of q, in the following. To get the mean squared displacement of the nanoparticle from an (experimentally recorded) incoherent intermediate scattering function, the following Gaussian approximation is often used:
Sα (t) = ce−(t/τ ) .
FIG. 12. Incoherent intermediate scattering function of the nano-particle at different temperatures with q = 7 (solid lines) and q = 1 (dashed lines).
10-2 10
-4
10-2
10-1
100
101
102
103
104
t FIG. 13. Directly calculated MSD (solid lines) and MSD calculated from Eq. (24) (dotted lines) for q = 7.
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124907-9
A. Kuhnhold and W. Paul
J. Chem. Phys. 141, 124907 (2014)
TABLE IV. Fit parameter for the stretched exponential fit (Eq. (25)) of the α-relaxation in the mean squared displacement of the embedded particle at different q at T = 0.6 in the range −6 ln (0.9)/q2 ≤ r(t)2 ≤ −6 ln (0.2)/q2 . q 0.60 0.50 0.40 0.30 0.20 0.10
β 1.00 ± 0.02 1.01 ± 0.03 0.97 ± 0.03 0.98 ± 0.03 1.1 ± 0.1 1.1 ± 0.3
τ 13 600 ± 100 19 600 ± 100 30 700 ± 200 55 900 ± 300 108 000 ± 1000 320 000 ± 10 000
additive term in (25) vanishes. To be better comparable to the results in Refs. 16 and 17, we fit the MSD for q < 0.65. Our results for T = 0.6 (33% above T*) are shown in Table IV. We find stretching exponents of about one to slightly above one for the smallest q-values. Again, we would conclude, that our results give no indication of a superdiffusive motion of the nano-particles at temperatures close to Tg . This is clearly so for q ≥ 0.3, but for q ≤ 0.2 we are susceptible to systematic uncertainties in the MSD used to predict the incoherent scattering. In the experiment, other factors may explain the finding of apparent superdiffusive motion, which is clearly incompatible with all known phenomenology of the glass transition and with our findings in this study in particular. We cannot reasonably speculate about these factors, but we think that in view of the disagreement with all other information available from experiment or simulation, the validity of the conclusion about superdiffusive motion on the basis of the XPCS experiments should be critically reexamined. V. CONCLUSIONS
We have presented a Molecular Dynamics simulation study on the temperature dependence of the motion of a nanoparticle suspended in a polymer melt and the applicability of the micro-rheological approach to the determination of the shear moduli of the polymer melt at low temperatures. The model system consists of short (N = 10) bead-spring polymer chains and is well studied in the literature.15, 19, 26 Following the suggestions of Vladkov and Barrat19 to calculate the complex shear modulus from the stress tensor autocorrelation and the Rouse modes, we showed the temperature dependence of the storage and loss moduli of our system. We found the typical properties of extending β- and α-relaxations with decreasing temperature, which means developing plateaus in the storage modulus and corresponding maxima in the loss modulus. The terms α- and β-relaxation correspond to global, structural and local rearrangements, respectively. The latter are related to the so-called caging effect, where a monomer is trapped by its neighbors. After this we studied the temperature dependent thermal motion of the suspended nano-particle, which also showed a caging effect represented by a plateau in the mean squared displacement. The decrease of the diffusion constant revealed the critical temperature of mode-coupling-theory for our model.26 From the MSD we calculated the complex modulus via the micro-rheological approach and found again prop-
erties of extended relaxations, but occurring at different frequencies compared to the melt moduli. A comparison of the zero-shear viscosities from micro-rheology and stress tensor calculation showed that the micro-rheological approach does not work for T < 0.6. This was explained by the fact that the particle does not explore the bulk, but collectively moves with the cage of particles surrounding its original position. This was confirmed by looking at the cooperative motion of the nano-particle and surrounding monomers in time and space, where we found a correlation in time over three decades and a strong layering effect in space for temperatures lower than or equal to 0.6. For such a case the theory of micro-rheology needs to be adapted, maybe via including the caging effect in an explicit way. In the last part we dealt with the incoherent intermediate scattering function, which showed a two-step decay for large enough q for both monomers and the nano-particle. The relaxation around the plateau is described by the β-relaxation and the long-time decay by the α-relaxation. The latter can be fitted with a stretched exponential function. We have fitted this function for various values of the scattering vector q, using the similarity of the incoherent scattering function with the mean squared displacement. The stretching exponent stayed in all cases around 1 in contrast to experimental findings of XPCS measurements,16, 17 which yielded a superdiffusive motion of the suspended particle at long times. Because of the restrictions of time scales which can be covered in numerical simulations, we cannot fully disprove those findings, but we wanted to point out that they need reexamination in view of all results on the behavior of relaxation functions in a glass forming system. 1 T.
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