Large and Long-Range Dynamic Correlations in Supercooled Fluids

Nov 17, 2014 - Revealed via Four-Point Correlation Functions ... one commonly used reveals the existence of large and long-range dynamic correlations,...
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Large and Long-Range Dynamic Correlations in Supercooled Fluids Revealed via Four-Point Correlation Functions Elijah Flenner* and Grzegorz Szamel* Department of Chemistry, Colorado State University, Fort Collins, Colorado 80523, United States ABSTRACT: Dynamic heterogeneity in supercooled fluids is often monitored using fourpoint structure factors. We show that a four-point structure factor slightly different from the one commonly used reveals the existence of large and long-range dynamic correlations, which dominate the correlations identified with dynamic heterogeneity. Specifically, we compare two four-point structure factors that differ by the mobility field used in their definition. The mobility field used in our alternative four-point structure factor, which we denote as SF4 (q, k; t), is the full microscopic self-intermediate scattering function, whereas the mobility field in the structure factor commonly used to examine dynamic heterogeneity, which we denote as Scos 4 (q, k; t), is is the real part of the microscopic self-intermediate scattering function. The susceptibility χF4(t) and the correlation length ξF4 (t) associated with SF4 (q, k; t) are both much larger than the susceptibility χcos 4 (t) and the correlation length cos F ξcos 4 (t) associated with S4 (q, k; t). In particular, we find that ξ4 (τα) grows proportionally with the relaxation time, ξF4(τα) ∼ τα, while we previously established that ξcos 4 (τα) has a much weaker dependence on the relaxation time. We argue that the growth of ξF4(τα) is primarily due to the growth of a transient elastic response, while the growth of ξcos 4 (τα) is due to spatially correlated domains of slow and fast particles, usually referred to as dynamic heterogeneity.

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INTRODUCTION Dynamic correlations are at the heart of many glass transition theories.1 An early theory of Adam and Gibbs2 postulated that upon supercooling increasingly larger domains of particles must cooperatively rearrange for structural relaxation to occur, and that the increase in the size of these domains is responsible for the dramatic increase of the relaxation time of supercooled liquids. Similarly, there is an increase in the size of dynamically correlated domains in random first order transition theory.3 While these theories are based on correlated motion leading to relaxation, within the facilitation picture4 relaxation is local, but dynamic correlations arise since particles cannot relax unless there is a neighboring relaxation event. Four-point structure factors are a convenient tool to examine and quantify dynamic correlations.5−7 To construct a four-point structure factor, one first defines a measure of mobility gn(t) that is sensitive to displacement δrn(t) of particle n over time interval t. The spatial correlations of mobility, where mobility is defined through the function gn(t), are then studied by examining a four-point structure factor S4g (q; t ) =

1 ⟨∑ g (t )gm*(t )e−iq·[rn(0) − rm(0)]⟩ N n,m n

To study particles that move slowly over a time t, one often used mobility function gn(t) is the overlap function ϕn(l; t),5,6,8−12 which is 1 if particle n has moved less than a distance l over a time t and 0 otherwise; ϕn(l; t) = θ(l − |δrn(t)|), where θ(·) is the Heaviside step function. In other words, ϕn(l; t) selects particles that have moved less than a distance l over a time t. With an appropriate choice of l, ϕn(l; t) can be used to measure the size of clusters of particles that are nearly stationary over a time t. Another common choice of gn(t) is the real part of the microscopic self-intermediate scattering function, cos{k·[rn(t) − rn(0)]},12−15 which is used to study dynamic correlations over a length scale l ≈ 2π/|k|. The advantage of this choice is that it allows one to investigate the anisotropy of dynamic correlations. In particular, studies that have used the real part of the microscopic self-intermediate scattering function (among others) have revealed that spatially correlated dynamics depends on the relative direction of motion of two particles.12,14−20 While four-point structure factors defined in terms of the overlap function and the real part of the selfintermediate scattering function reveal slightly different features of correlated dynamics, the dynamic susceptibilities and correlation lengths obtained from these functions are comparable and follow similar relationships to the liquid’s relaxation time.12

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The overall strength of dynamic correlations (roughly corresponding to the average number of correlated particles) is given by the dynamic susceptibility χg4(t) = limq→0 Sg4(q; t), and the spatial extent of the correlations is given by the dynamic correlation length ξg4(t), which can be determined by analyzing the small q behavior of Sg4(q; t).7 © XXXX American Chemical Society

Special Issue: Branka M. Ladanyi Festschrift Received: September 17, 2014 Revised: November 14, 2014

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the production runs after equilibrating each run using a Nosé− Hoover NVT thermostat for at least 100τα. We performed the simulations using LAMMPS.23

In this work, we examine a slightly different four-point structure factor and the resulting dynamic susceptibility and correlation length. We choose gn(t) to be the full microscopic self-intermediate scattering function, including both the real and imaginary parts. Note that the thermodynamic average of the imaginary part vanishes, which could lead one to naively expect that the dynamic correlations revealed by this alternative structure factor should be similar to those discussed above. However, we show that the dynamic susceptibility χg4(t) and the correlation length ξg4(t) are much larger when the mobility function is chosen to be the full self-intermediate scattering function rather than the real part of the self-intermediate scattering function. The two correlation lengths have a very different temperature dependence, and thus, they correlate with the relaxation time τα in very different ways. We find that ξg4(τα) where gn(t) is the full self-intermediate scattering function is proportional to the relaxation time, while ξg4(τα) for gn(t) being the real part of the self-intermediate scattering function has a weaker (and more complicated) dependence on the relaxation time12 that is similar to the growth of ξg4(τα) when gn(t) is chosen to be the overlap function. The paper is organized as follows. In the next section, we describe the details of the simulations. Next, we examine fourpoint structure factors where gn(t) is chosen to be the microscopic self-intermediate scattering function and the real part of the microscopic self-intermediate scattering function, and show that these two choices lead to very different susceptibilities and correlation lengths. We examine the temperature and relaxation time dependence of the correlation lengths over the temperature range examined in this work. We finish with some concluding remarks. A detailed analysis of an independent evaluation of dynamic susceptibility (which avoids extrapolation of the four-point structure factor to zero wavevector) is presented in the Appendix.

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FOUR-POINT STRUCTURE FACTOR

In this section, we examine four-point structure factors S4g (q; t ) =

1 ⟨∑ g (t )gm*(t )e−iq·[rn(0) − rm(0)]⟩ N n,m n

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where we choose the weight function gn(t) to be the full microscopic self-intermediate scattering function, gn(t) = F̂n(k; t) = eik·[rn(t)−rn(0)], and the real part of the microscopic selfintermediate scattering function, gn(t) = cos{k·[rn(t) − rn(0)]}. The resulting four-point structure factors will depend on both q and k wave-vectors, and this dependence will be denoted as Sg4(q, k; t). We note that, in previous work,14 we used the opposite order of the arguments q and k in the labeling of S4. We emphasize that we keep the same meaning of the wavevectors: q is used to measure the extent of dynamic correlations, and k is used to define the particle mobility. To simplify the notation, we will not explicitly denote the k dependence of the weight function and the resulting susceptibility, and correlation length, except in the Appendix. To denote the structure factor defined in terms of F̂n and the resulting susceptibility and correlation length, we will use superscript F, and to denote the structure factor defined in terms of cos (i.e., in terms of the real part of F̂n) and the resulting susceptibility and correlation length, we will use superscript cos. We find that these two choices result in very different four-point structure factors, dynamic susceptibilities, and correlation lengths. Most importantly, χF4 and ξF4 are much cos larger than χcos 4 and ξ4 . First, we note that the thermodynamic averages of eik·[rn(t)−rn(0)] and cos{k·[rn(t) − rn(0)]} are identical, since

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SIMULATION DETAILS We studied a 50:50 mixture of repulsive harmonic spheres that has been extensively studied in the literature.10−12,21,22 The interaction potential is given by 2 ε⎛ r ⎞ Vnm(r ) = ⎜1 − ⎟ σnm ⎠ 2⎝

=0

if

r > σnm

if

r ≤ σnm

(2) (3)

The parameters are σ22 = 1.4σ11 and σ12 = 1.2σ11, and these parameters are chosen to inhibit crystallization. We present our results in reduced units with σ11 being the unit of length, (mσ112/ε)1/2 the unit of time, and 10−4ε the unit of temperature. We examined the system at a number density ρ = N/V = 0.675 and temperatures T = 25, 20, 15, 2, 10, 9, and 8. To examine the small q behavior of the four-point structure factor, we examined very large systems consisting of N = N1 + N2 = 800 000 at all temperatures. Additionally, we studied an even larger system consisting of N = 6.4 million particles at T = 15. We also used trajectories from earlier simulational studies11,12 which used systems consisting of N = 10 000 and N = 100 000 particles. These smaller trajectories are used to obtain accurate susceptibilities. For reference, the onset of supercooling is around T = 12 and the mode-coupling temperature is around T = 5.2. We performed NVE simulations using the velocity Verlet algorithm with a time step of 0.02 for

Figure 1. Self-intermediate scattering function for T = 25, 20, 15, 12, 10, 9, and 8 (solid lines) listed from left to right. At the lower temperatures, a plateau begins to emerge.

⟨sin{k·[rn(t) − rn(0)]}⟩ = 0. In Figure 1, we show this average, the self-intermediate scattering function 1 Fs(k ; t ) = ⟨∑ ei k·[rn(t ) − rn(0)]⟩ N n (5) for temperatures corresponding to the normal liquid at T = 25 to a temperature of T = 8, which is slightly below the onset of supercooling at T = 12. B

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At the lower temperatures, there is an initial short time decay followed by the final α relaxation. We note that a plateau begins to emerge between the initial decay and the α relaxation. The α relaxation can be fit to a stretched exponential, Ae−(t/τ)β, and a plateau height A and stretching exponent β can determined from these fits. For this temperature range, the plateau height is A ≈ 0.7 and depends little on temperature. The stretching exponent β varies from 0.8 at the highest temperature to 0.7 at the lowest temperature. We define τα as when Fs(k; τα) = e−1, which is not the τ found from the stretched exponential fits. The temperature dependence of these fits is described in detail in ref 11. As we discussed in ref 12, both choices of g(t) result in an anisotropic four-point structure factor with correlation lengths depending on the angle between q and k. However, the q → 0 limits of SF4 (q, k; t) and Scos 4 (q, k; t), while different, do not depend on the angle between q and k. For this study, we fix |k| = 6.1, which is around the first peak of the static structure factor, and examine two relative orientations of q and k: the case when q is perpendicular and parallel to k. Both the susceptibilities and correlation lengths depend on k. However, as we mentioned above, to simplify notation, we will only indicate their dependence on t. Shown in Figure 2 is SF4(q, k; τα) (circles) and Scos 4 (q, k; τα) (squares) for q perpendicular (open symbols) and parallel (closed symbols) to k at T = 25, 15, 12, and 8 for systems of 800 000 particles. The four-point structure factors are strikingly different for these two choices of gn(t). Note the dramatic upturn for small wave-vectors can be seen for T = 15 and 12, indicating a large correlation length when q and k are parallel and gn(t) = F̂s(k; t). We believe that a similar upturn would be visible in the T = 8 data at wave-vectors smaller than those accessible in the present study. If we only examined q ≥ 0.2, the range examined in most simulations, for T = 15, 12, and 8, it would appear that there are different susceptibilities for q parallel to k and q perpendicular to k. Thus, the true small q behavior of four-point structure factors may be hidden for some measures of mobility and care must be taken when extrapolating Sg4(q, k; t) to q = 0. It was this observation that led us to perform simulations of very large systems consisting of 800 000 particles so we can reliably examine the extrapolation of Sg4(q, k; t) to q = 0 for a range of temperatures. It is evident from these large simulations that for T ≥ 12 the q → 0 limit of SF4(q, k; τα) is the same for q and k parallel and perpendicular. However, we would need even larger systems to extrapolate SF4(q, k; τα) for T = 10 and below. We also note that a plateau forms for T = 12 around q = 0.2, which could lead one to believe that one could accurately extrapolate SF4(q, k; τα) to q → 0. An extrapolation without wave-vectors smaller than 0.2 would result in a dynamic susceptibility much smaller than its actual value. We turn to the examination of the dynamic correlation lengths ξF4 (t) and ξcos 4 (t), which we find by fitting the small wave-vector dependence of the four-point structure factors to an Ornstein−Zernike function S4F(q, k; t ) ≈

Figure 2. Four-point structure factors SF4 (q, k; τα) (circles) and Scos 4 (q, k; τα) (squares) for k parallel to q (closed symbols) and k perpendicular to q (open symbols) for T = 25, 15, 12, and 8. Larger systems would be needed to obtain correlation lengths for q and k parallel at T = 8.

calculate the susceptibilities are presented in the Appendix. We fit the range q ≤ 0.12 for q and k parallel and the range q ≤ 0.2 for q and k perpendicular. We first examine the correlation lengths at the α relaxation time. Shown in Figure 3 are ξF4 (τα) (circles) and ξcos 4 (τα) (squares) for q parallel to k (closed symbols) and q perpendicular to k (open symbols) as a function of temperature T. For T ≤ 10, we cannot fit SF4(q, k; τα) when q and k are parallel even with systems of 800 000 particles. The lengths are always larger when k and q are parallel than perpendicular, which indicates the longer range of correlations for particles moving in the direction parallel to their separation vector versus in the direction perpendicular to their separation vector. For the larger temperatures, the temperature dependence of ξcos 4 (τα) and ξF4(τα) is not noticeably different, but there is a dramatic upturn in the temperature dependence for ξF4 (τα) around the onset of supercooling at T = 12. Different temperature dependences of ξF4(τα) and ξcos 4 (τα) lead to different correlations between these lengths and relaxation times. To show this explicitly, we plot ξF4 (τα) and F ξcos 4 (τα) versus τα in Figure 4. We note that the ξ4 (τα) and ξcos (τ ) appear to have a similar relationship to τ for higher 4 α α

χ4F (t ) 1 + (ξ4F(t )q)2

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and similarly for Scos 4 (q, k; t). The fitting is facilitated by using dynamic susceptibilities χF4(t) and χcos 4 (t) which are calculated independently, using a version of the procedure originally proposed by Berthier et al.24 The details of the method used to C

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Figure 3. Dynamic correlation lengths ξF4(τα) (circles) and ξcos 4 (τα) (squares) as a function of temperature for q and k parallel (closed symbols) and perpendicular (open symbols). The correlation length ξF4(τα) increases much faster than ξcos 4 (τα) with decreasing temperature for T ≤ 12, i.e., below the onset of supercooling. Our systems are not large enough to obtain correlation lengths for q parallel to k for T ≤ 10.

Figure 5. Four-point structure factor SF4(q, k; t) for T = 15 shown for τα, 2τα, 4τα, and 6τα calculated for a system of 6.4 million particles. The closed symbols are for q parallel to k, and the open symbols are for q perpendicular to k. The symbols at q = 0 are the susceptibilities calculated using the procedure described in the Appendix.

at q = 0 are from the independent calculations of the susceptibilities (the details of these calculations are given in the Appendix). We obtained dynamic correlation lengths by fitting fourpoint structure factors to the Ornstein−Zernike form as described above. Shown in Figure 6 are ξcos 4 (τα) (squares)

Figure 4. Dynamic correlation lengths ξF4(τα) (circles) and ξcos 4 (τα) (squares) as a function of τα for q perpendicular to k (open symbols) and q parallel to k (closed symbols). The lines are fits of ξF4(τα) to ξF4(τα) = aτα.

temperatures, but there is a very different relationship for temperatures below the onset of supercooling. We find that ξF4(τα) = aτα (solid lines) gives a reasonable fit to the full range of the data. The fit parameters were a ≈ 0.054 for q perpendicular to k and a ≈ 0.170 for q parallel to k. We extensively examined the relationship between ξcos 4 (τα) and τα in ref 12, so we do not repeat that analysis here. We note that a linear relationship between the maximum length that supports a hydrodynamic transverse wave mode, S t,25 and a relaxation time is predicted by a general constitutive model for supercooled liquids proposed by Mizuno and Yamamoto.26 For this model, S t = 2ctτs0, where τs0 is on the order of the structural relaxation time. This opens a possibility of a relation between ξF4(τα) and the hydrodynamic length S t. However, it is important to note that ξF4(t) also depends on time, while S t is unique, independent of time length. To examine the time dependence of the correlation lengths, we performed simulations of 6.4 million particles at T = 15. This was the lowest temperature where by running simulations over a few weeks time we were able to confirm that q → 0 extrapolations agreed with the calculated values of χF4(t) and we could obtain reasonable fits to SF4(q, k; t) for times up to several τα. Shown in Figure 5 are SF4(q, k; t) for several multiples of τα calculated using simulations of 6.4 million particles. The points

Figure 6. Dynamic correlation lengths ξF4(t) (circles) and ξcos 4 (t) (squares) when k and q are parallel (closed symbols) and when k and q are perpendicular (open symbols) calculated for T = 15 at several different times.

and ξF4 (τα) (circles) for q parallel to k (closed symbols) and q perpendicular to k (open symbols) for T = 15 for up to 7τα. At all times, the length for k and q parallel is larger than the one for k and q perpendicular. The dynamic correlation lengths ξF4 grow in time until about 5τα and then plateau. The correlation lengths ξcos 4 (t) grow until around 2τα and then plateau. F Furthermore, ξcos 4 (t) is smaller than ξ4 (t) at each time that we studied. For the time range that we can calculate correlation lengths, we do not observe a statistically significant decrease of their values, but we also cannot rule out such a decrease at later times. The plateau value for ξF4(t) for q and k parallel is around 30 particle diameters, while it is about 6 particle diameters for q and k perpendicular. This contrasts the much smaller values of 2.25 when q and k are parallel and 1.4 particle diameters when q and k are perpendicular for ξcos 4 (t). We note that the maximum in the correlation lengths does not occur at τα for ξF4 (t) or ξcos 4 (t), and this needs further study. D

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four-point structure factor or calculate corrections to account for the suppressed fluctuations. We found in the “Four-Point Structure Factor” section that the extrapolation of four-point structure factors has to be taken with care and one needs to be sure to have a sufficient number of small wave-vectors. However, it is difficult to know what is small enough, and the calculation of the suppressed fluctuations helps one determine if the available wave-vector range includes a sufficient number of small wave-vectors. We begin with some background and outline how to calculate the dynamic susceptibilities. We break the corrections due to suppressed fluctuations into three parts: the first term is the susceptibility in the absence of dynamic correlations, the second term is proportional to derivatives of the average mobility function, and the third term is a consequence of total momentum conservation. This third term is usually not discussed, and we demonstrate that it dominates the susceptibility for some definitions of mobility. Next, we present results for two definitions of mobility used in this paper, the full microscopic self-intermediate scattering function and its real part. We show that the susceptibilites calculated using these two definitions at low temperatures differ by orders of magnitude. We should note here that Chandler and collaborators28 investigated earlier dynamic susceptibilities defined in terms of the full microscopic self-intermediate scattering function. However, when calculating contributions due to suppressed fluctuations, they omitted the term originating from the fluctuations of the total momentum. As a result, they could not see a huge difference between the susceptibilites calculated using two different definitions of mobility. We emphasize that without the term originating from the fluctuations of the total momentum we cannot explain the very large difference between SF(q, k; τα) and Scos(q, k; τα) that is evident already at relatively high temperatures, where extrapolation of the four-point structure factors to q = 0 is feasible; see Figure 2a.

In summary, there is a striking difference between the two four-point structure factors, SF4(q, k; t) and Scos 4 (q, k; t). They give a much different picture of spatially correlated dynamics. This difference is due to the different kinds of correlations which are revealed by these two structure factors, and different growth of these correlations upon approaching the glass transition.

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CONCLUSIONS We examined spatial correlations of two closely related measures of mobility: the full microscopic self-intermediate scattering function and the real part of the microscopic selfintermediate scattering function. These two measures of mobility have the same thermodynamic average, but at lower temperatures, their correlations differ by orders of magnitude. In particular, the dynamic susceptibility, a measure of the number of dynamically correlated particles, and the dynamic correlation length, a measure of the spatial extent of dynamically correlated regions, are very different. We believe that this behavior is exhibited in all systems with dynamics that conserves the total momentum and, in fact, we first noticed it in the Kob−Andersen Lennard-Jones binary mixture. Studying the four-point structure factors that measure correlations of mobility demonstrates that different dynamic correlations need to be considered in supercooled liquids. The differences in the correlations are evident when the relationship between the correlation lengths and the α relaxation time τα is examined. One length, ξF4 (τα), grows linearly with τα, while the other, ξcos 4 (τα), has a more complicated relationship that may have two logarithmic regimes.11,12 The linear growth with τα of ξ4F(τα) suggests that ξ4F(τα) is related to the maximum wavelength of a shear wave, S t,25,26 which would imply that the correlation length ξF4(τα) in the supercooled fluid is related to the growing elastic response. However, this interpretation raises several questions. One question is what is the physics that gives rise to the difference between ξF4(t) when q and k are parallel and perpendicular. It would be interesting to obtain the relationship between ξF4(τα) and S t to see if they are indeed related. Previous work demonstrated that ξcos 4 (t) is related to spatially correlated heterogeneous dynamics,12 i.e., the fact that some particles move much faster and much slower than would normally be expected for a Gaussian distribution of particle displacements. Moreover, this correlation length is also sensitive to the shape of the dynamically correlated regions and depends on the angle between q and k12,14,15 used in the definition of the four-point structure factor Scos 4 (q, k; t). In spite of some caveats mentioned above, we believe that the new length, ξF4(t), reflects the transient elasticity of the supercooled fluid. To prove this assertion, further simulation studies at lower temperatures, including in the glass, are needed. This work is under way and will be reported elsewhere.

Background

It is convenient for future discussion to consider a more general four-point structure susceptibility χg4(k1, k2; t) defined as the q → 0 limit of the following (also more general) four-point structure factor Sg4(q, k1, k2; t) χ4 (k1, k 2; t ) =

1 ⟨∑ g (k1; t )gm*(k 2; t )⟩ N n,m n −

S4g (q, k1, k 2; t ) =

1 ⟨∑ g (k1; t )⟩⟨∑ gm*(k 2; t )⟩ N n n m

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1 ⟨∑ g (k1; t )gm*(k 2; t )e−iq·[rn(0) − rm(0)]⟩ N n,m n (8)

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In the above formulas, we explicitly denoted the wave-vector dependence of the weight function gn(k; t). Also, in the above formulas and in the rest of the Appendix, we explicitly denote the wave-vector dependence of the suceptibility and correlation length. Most studies examine the case of k1 = k2, but we will not make this restriction in this appendix. However, we will restrict ourselves to the case when |k1| = |k2|. Furthermore, note that, since we put q = 0, we need to include the subtraction term, the second term on the right-hand side of eq 7. Considerations of the transformations between ensembles helps one understand the correlations examined by Sg4(q, k1, k2;

APPENDIX: DYNAMIC SUSCEPTIBILITY Dynamic susceptibilities provide insight into dynamic correlations in fluids and glasses and allow for the critical examination of glass transition theories.7,27 Since a dynamic susceptibility is the fluctuation of a total mobility (i.e., of the sum of the mobilities pertaining to individual particles) and some global fluctuations are suppressed in the simulation ensembles used for studies of supercooled liquids, to find the ensemble independent susceptibility, one has to either extrapolate the E

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t). Some global fluctuations are suppressed in specific ensembles, and χ4(k1, k2; t) calculated directly from a simulation using eq 7 differs from limq→0 S4(q, k1, k2; t). There are two possibilities to obtain an ensemble independent χ4(k1, k2; t): to extrapolate S4(q, k1, k2; t), which may require very small wave-vectors and very large systems, or to calculate the fluctuations that are suppressed in the given ensemble.10,15,29 For the case of the NVE simulational ensemble, the second procedure results in the following expression for the total susceptibility χ4g (k1, k 2; t ) = χ4g (k1, k 2; t )|NVE + ⟨g (k , t )⟩2 S(q = 0) + Xδgg (k1 , k 2 ; t ) +

Xδgp(k1,

k 2; t )

Figure 7. Four-point susceptibility χF4(k, k; t) for T = 8, 10, 12, 15, 20, and 25 (solid lines) listed from top to bottom and XFδp(k, k; t) (dashed lines) for the same temperatures shown on a log−log scale. The dotted line is χcos 4 (k, k; t) for T = 8, which is almost 3 orders of magnitude smaller than χF4(k, k; t) at this mildly supercooled temperature. The XFδp(k, k; t) term dominates the susceptibility for temperatures below the onset of supercooling, and is a good approximation for χF4(k, k; t) for these temperatures. The open circles on T = 15 are the extrapolations of SF4 (q, k, k; t) for q and k perpendicular.

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χg4(k1,

where k2; t)|NVE is the susceptibility calculated directly from an NVE simulation, Xgδg(k1, k1; t) can be expressed in terms of the derivatives of ⟨gn(k; t)⟩ with respect to temperature, density, and concentration, and Xgδp(k1, k2; t) is due to fluctuations of the total momentum, which is also fixed in the simulational NVE ensemble. In eq 9 S(q) =

1 ⟨∑ e−iq·[rn(0) − rm(0)]⟩ N n,m

The symbols on the T = 15 curve are the extrapolated values from SF4(q, k, k; t) (which is characterized in the “Four-Point Structure Factor” section) where we extrapolated the data when q and k are perpendicular. The contribution due to Xgδp(k, k; t) becomes a good approximation to χF4(k, k; t) for T ≤ 12, i.e., for temperatures below the onset of supercooling. To understand why the susceptibilities χF4(k, k; t) and χcos 4 (k, k; t) are so different, first note that XFδp(k, k; t) = (kBT/m) F k2t2Fs2(k; t). Then, notice that Scos 4 (q, k, k; t) = (1/4)[S4 (q, k, F F F k; t) + S4 (q, k, −k; t) + S4(q, −k,k; t) + S4(q, −k, −k; t)], which leads to four Xgδp correction terms that sum to zero. We note that when the mobility function is chosen to be the overlap function, θ(l; t), the contribution due to Xgδp is zero, since the correction can be written as an integral over k1·k2 weighted by functions that only depend on the magnitude of k1 and k2. We also note that χF4(k, k; t)|NVE and χcos 4 (k, k; t)|NVE are also different, which is easily seen by comparing the limits when t goes to infinity. In this limit, χF4(k, k; ∞)|NVE = 1 and χcos 4 (k, k; ∞)|NVE = 1/2, since the average of cos2(x) and sin2(x) are both 1/2. In contrast, it is important to note that XFδp(k, k; t) is equal to Xcos δp (k, k; t). To contrast very large susceptibilities χF4(k, k; t) and much smaller values of χcos 4 (k, k; t), we show the latter quantity in Figure 8. Note that the y-axis is linear instead of logarithmic, as in the figure showing χF4(k, k; t), Figure 7. For T = 8 and 20, we show χcos 4 (k, k; t)|NVE as dashed lines. To emphasize the large difference between the susceptibilities at low temperatures, we cos show χcos 4 (k, k; t) in Figure 7 for T = 8 as the dotted line; χ4 (k, F k; t) is almost 3 orders of magnitude smaller than χ4(k, k; t) for this temperature. One might naively expect the dynamic susceptibilities constructed from two functions that have the same thermodynamic average to be similar and not differ by orders of magnitude. However, we have shown that, if one considers the full microscopic intermediate scattering function or the real part of this function, the susceptibility does differ by orders of magnitude. This is due to the full microscopic intermediate scattering containing information that is sensitive to an emerging elastic response.

(10)

is the static structure factor. The ⟨g(k; t)⟩2S(q = 0) term is the q → 0 limit of S4(q, k1, k2; t) when there are no dynamic correlations. The contributions to the Xgδg(k1, k2; t) term have been discussed in the literature.30,31 They can be related to parts of the local energy, density, concentration, etc., that are correlated with the fluctuation of the local mobility defined through g(k; t). It is important to recognize that Xgδg(t) does not capture all of the suppressed fluctuations. As we already stated above, in an NVE simulation, the total momentum is also conserved, and if one considers fluctuations of the total momentum in the same manner as other conserved quantities, one finds that there is an additional correction to χg4(k1, k2; t),29 which is given by Xδgp(k1, k 2; t ) =

kBT k1·k 2t 2⟨g (k1; t )⟩⟨g *(k 2; t )⟩ m

(11)

In general, Xgδp(k1, k2; t) depends on the angle between k1 and k2 (note that Xgδg(k1, k2; t) does not depend on the angle between k1 and k2). In particular, if one were to choose k1 perpendicular to k2, then Xgδp(k1, k2; t) is zero, but Xgδp(k1, k2; t) is nonzero if k1 is parallel to k2. Furthermore, Xgδp(k1, k2; t) calculated at the structural relaxation time of the fluid, τα, is going to show a very different temperature dependence than Xgδp(k1, k2; t) calculated at τα. Susceptibilities χF4(k; t) and χcos 4 (k; t)

One natural choice of mobility function gn(t) is the microscopic self-intermediate scattering function F̂n(k; t) = e−ik·[rn(t)−rn(0)]. Previous works typically only examine correlations of the real part of F̂n(k; t) = cos{k·[rn(t) − rn(0)]}, and we show that choosing the full microscopic self-intermediate scattering function leads to a very different dynamic susceptibility than choosing only the real part. While the thermodynamic averages of these two choices of the mobility are identical, the fluctuations are very different. In Figure 7, we show as solid lines χF4(k, k; t) calculated using eq 9 and as dashed lines Xgδp(k, k; t) for T = 8, 10, 12, 15, 20, and 25. F

dx.doi.org/10.1021/jp509442a | J. Phys. Chem. B XXXX, XXX, XXX−XXX

The Journal of Physical Chemistry B

Article

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Figure 8. Four-point susceptibility χcos 4 (k, k; t) for T = 8, 10, 12, 15, 20, and 25 (solid lines) listed from top to bottom. The dotted lines are cos χcos 4 (k, k; t)|NVE for T = 8 and T = 20. The Xδp (k, k; t) terms are an cos increasing larger contribution to χ4 (k, k; t) with decreasing temperature.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: fl[email protected]. *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We gratefully acknowledge the support of NSF Grant No. CHE 1213401. This research utilized the CSU ISTeC Cray HPC System supported by NSF Grant CNS-0923386.

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REFERENCES

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dx.doi.org/10.1021/jp509442a | J. Phys. Chem. B XXXX, XXX, XXX−XXX