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Transport dynamics of complex fluids
Edited by Steve Granick, Institute for Basic Science Center for Soft and Living Matter, Ulju-gun, Ulsan, Republic of Korea, and approved May 9, 2019 (received for review January 7, 2019)

Significance
Many disordered fluid systems exhibit anomalous transport dynamics, which do not obey Einstein's theory of Brownian motion or other currently available theories. Here, we present a new transport equation governing thermal motion of complex fluidic systems, which provides a unified, quantitative explanation of the mean-square displacement, the non-Gaussian parameter, and the displacement distribution of various complex fluids. The applicability of our theory is demonstrated for molecular dynamics simulation results of supercooled water and dense hard disc fluids and for experimental results of colloidal beads diffusing on lipid tubes. This work suggests previously unexplored directions for quantitative investigation into transport and transport-coupled processes in complex disordered media, including living cells.
Abstract
Thermal motion in complex fluids is a complicated stochastic process but ubiquitously exhibits initial ballistic, intermediate subdiffusive, and long-time diffusive motion, unless interrupted. Despite its relevance to numerous dynamical processes of interest in modern science, a unified, quantitative understanding of thermal motion in complex fluids remains a challenging problem. Here, we present a transport equation and its solutions, which yield a unified quantitative explanation of the mean-square displacement (MSD), the non-Gaussian parameter (NGP), and the displacement distribution of complex fluids. In our approach, the environment-coupled diffusion kernel and its time correlation function (TCF) are the essential quantities that determine transport dynamics and characterize mobility fluctuation of complex fluids; their time profiles are directly extractable from a model-free analysis of the MSD and NGP or, with greater computational expense, from the two-point and four-point velocity autocorrelation functions. We construct a general, explicit model of the diffusion kernel, comprising one unbound-mode and multiple bound-mode components, which provides an excellent approximate description of transport dynamics of various complex fluidic systems such as supercooled water, colloidal beads diffusing on lipid tubes, and dense hard disk fluid. We also introduce the concepts of intrinsic disorder and extrinsic disorder that have distinct effects on transport dynamics and different dependencies on temperature and density. This work presents an unexplored direction for quantitative understanding of transport and transport-coupled processes in complex disordered media.
- complex fluids
- thermal motion
- diffusion kernel correlation
- supercooled water
- colloidal particles on lipid tube
Thermal motion in complex fluids is a complex stochastic process, which underlies a diverse range of dynamical processes of interest in modern science. Since Einstein’s seminal work on Brownian motion (1), thermal motion in condensed media has been the subject of a great deal of research. However, it is still challenging to achieve a quantitative understanding of the transport dynamics of disordered fluidic systems, including cell nuclei and cytosols (2), membranes and biological tissue (3), polymeric fluid (4), supercooled water (5, 6), ionic liquids (7, 8), and dense hard-disk fluids (9). Interestingly, thermal motion of these complex fluids commonly exhibits a mean-square displacement (MSD) with initial ballistic, intermediate subdiffusive, and then terminal diffusive behavior (10⇓–12); the associated displacement distribution is non-Gaussian except in the short- and long-time limits, and its deviation from Gaussian increases at short times but decreases at long times, as long as thermal motion is uninterrupted. These phenomena cannot be quantitatively explained by the original theory of Brownian motion (13).
To explain dynamics of anomalous thermal motion, the theory of Brownian motion has undergone a number of generalizations (14⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–27). Among these generalizations, Montroll and Weiss’s (20) continuous-time random walk (CTRW) model enables quantitative description of anomalous transport caused by a tracer particle being trapped by other objects in disordered solid media (28); the CTRW with a power-law waiting-time distribution (WTD),
While these models successfully describe subdiffusive transport occurring in disordered media, a number of disordered fluidic systems exhibit Fickian yet non-Gaussian diffusion (36⇓⇓⇓⇓⇓⇓–43); that is, the MSD is linearly proportional to time but the displacement distribution deviates from Gaussian. This issue was recently addressed by stochastic diffusivity (SD) models, in which the diffusion coefficient is treated as a stochastic variable (44⇓⇓⇓⇓⇓⇓–51). While SD models successfully demonstrate Fickian yet non-Gaussian diffusion, to the best of our knowledge, these models, too, are inconsistent with the transient subdiffusive MSD and the nonmonotonic time dependence of the non-Gaussian parameter (NGP), widely observed features of complex fluids.
The NGP has a long history in transport theory. Rahman, Singwi, and Sjölander first noted that the NGP is an observable in neutron scattering experiments (52), and Rahman recognized it as the first-order coefficient in the Hermite-polynomial expansion of the displacement distribution around Gaussian (53). Nieuwenhuizen and Ernst (54) showed that the NGP, or the fourth cumulant of displacement, is related to the time-correlation function (TCF) of the diffusion coefficient fluctuation and the Burnett correlation function (BCF), a functional of velocity autocorrelation functions (VAFs), for a system of independent charged particles hopping on a one-dimensional lattice with static disorder. Later, the NGP and BCF were investigated for interacting gas and fluid systems (55, 56) and, more recently, also for glassy systems (57, 58) for which the authors suggested the NGP as a measure of the diffusion coefficient fluctuation and dynamic heterogeneity. However, for complex fluid systems commonly exhibiting initial ballistic and intermediate subdiffusive thermal motion before terminal diffusion, the TCF of the diffusion coefficient is neither well defined nor a good measure of mobility fluctuation before terminal diffusion emerges. For complex fluid systems exhibiting non-Fickian diffusion, there is no precise definition of mobility fluctuation or exact relationship of mobility fluctuation with the NGP and VAFs. Of course, it remains a challenge to achieve a unified, quantitative understanding of the anomalous MSD, nonmonotonic NGP, and non-Gaussian displacement distribution of various complex fluid systems.
Transport Equation of Complex Fluids
Here, to address these issues, we present a transport equation that provides a quantitative description of thermal motion for various complex fluids. This equation can be derived by considering the continuum limit of a random walk model with a general sojourn time distribution,
Schematic representation of our random walk model with environmental state-dependent dynamics. In our model, the sojourn time distribution
For the sake of generality, we do not assume a particular model of environmental state dynamics, nor do we assume a particular form of mathematical operator
Eq. 2 encompasses the CTRW model and the diffusing diffusivity models (45⇓⇓⇓⇓⇓–51) (see Discussion for more details). A further generalization of Eq. 2 for complex fluidic systems under a spatially heterogeneous external potential is presented in SI Appendix, Text S2.
Analytic Expressions of the Moments
From Eq. 2, we obtain the exact analytical expressions of the first two nonvanishing moments,
The mean diffusion kernel,
While the second moment,
Model-based quantitative analysis of the mean-square displacement (MSD) and non-Gaussian parameter (NGP) for the TIP4P/2005 water system. (A and B) Time profiles of the MSD and NGP at various temperatures. Open circles, simulation results; solid lines, the best fits of Eqs. 9 and 18 to the simulation results. (C) Mean diffusion kernel obtained from Eq. 10 with optimized parameter values. (D) Solid lines, diffusion kernel correlation,
In our theory,
Using Eqs. 3a and 3b and the definition of the NGP,
Model-free quantitative analysis of the mean-square displacement (MSD) and non-Gaussian parameter (NGP) for the TIP4P/2005 water system at 193 K. (A and B) Time profiles of the MSD and NGP: circles, simulation results; blue lines, best fits of Eqs. 9 and 18. (C) Mean diffusion kernel: circles, two-point VAF obtained from simulation; red line, second-order time derivative of MSD data; blue line, Eq. 10 with optimized parameters given in Table 1. (D) Time profile of diffusion kernel correlation extracted by analyzing the MSD and NGP data using (red line) model-free theory, (blue line) model-based theory, and (yellow dashed-dotted line) contribution of the unbound mode. (D, Inset) Time profile of
Diffusion Kernel Correlation and Velocity Autocorrelation Functions
The diffusion kernel correlation,
Microscopic measurement of the bound- and unbound-mode components of diffusion kernel correlation for the TIP4P/2005 water system at 193 K. (A) Burnett correlation function (BCF): red circles,
Ergodicity and Long-Time Limit of Diffusion Kernel Correlation and Non-Gaussian Parameter
For fluidic systems showing terminal Fickian diffusion,
Intrinsic and Extrinsic Disorder
For ergodic fluid systems, the long-time limit value of the product between the MSD and NGP serves as a measure of disorder strength. This disorder strength measure is decomposable into intrinsic and extrinsic disorder. To show this, consider the long-time asymptotic behavior of the MSD and NGP:
From Eqs. 7a and 7b, we obtain the long-time limit value of the product of the MSD and NGP as
Intrinsic disorder is far less sensitive to the temperature and density of media than extrinsic disorder and can be easily estimated from Eq. 7a, the asymptotic long-time behavior of the MSD. Extrinsic disorder can be estimated by a direct numerical calculation of
Disorder strength, defined in Eq. 8, is directly related to the whole-time integration of the BCF; that is,
Model-Based Quantitative Analysis of the MSD and NGP
Intrinsic disorder causes the MSD to deviate from
The applicability of Eq. 9 to various disordered fluid systems implies a universality in the MSD time profile of disordered fluids, which is decomposable into one unbound-mode dynamic and multiple bound-mode dynamics, comparable to viscoelastic motion of a bead in a polymer network. At short times, a tracer molecule is trapped by the surrounding molecules. This bound state consists of multiple bound modes, each with their own characteristic frequencies. Meanwhile, at long times, a tracer molecule escapes the cage of the surrounding molecules and moves around in the media, repeatedly being caged and escaping the cage. The first term on the right-hand side (R.H.S.) of Eq. 9 accounts for the unbound mode, and the second term accounts for the bound modes. In Eq. 9,
At all times, temperatures, and densities investigated, Eq. 9 with only two bound modes
The NGP is dependent not only on the mean transport dynamics,
The NGP of disordered fluids is a nonmonotonic function of time with a single peak. According to our model, the NGP quadratically increases with time,
It was recently shown that diffusion coefficient fluctuation strongly correlates with string-like cooperative motion in dense fluids (57, 58), which is reportedly related to the NGP peak height (58). We find that the NGP peak height,
Explicit Model for Diffusion Kernel
In the previous section, we demonstrated that the time profile of
Optimized values of adjustable parameters for supercooled water at 193 K
We note here that the whole-time integration of the diffusion kernel,
Model-Free Quantitative Analysis of the MSD and NGP
By analyzing the numerical data of the time-dependent MSD and NGP using Eqs. 3a and 3b, we can extract the time profiles of the mean diffusion kernel and the DKC without assuming a physical model, such as Eq. 9. In Fig. 3, we demonstrate this model-free analysis of the MSD and NGP data for supercooled water at 193 K (Methods).
According to our simulation, shown in Fig. 3A, the MSD of supercooled water exhibits oscillatory behavior with a slight bump and dip between 0.1 ps and 1 ps. This mysterious nonmonotonic oscillation in the MSD time profile of supercooled water was previously reported in the literature (5, 68, 69). We find the nonmonotonic MSD time dependence is unrelated to the finite-size effect and emerges not only under a constant temperature condition but also under the constant energy condition (SI Appendix, Fig. S7). The origin of the slight oscillation in the MSD time profile may be attributable to the intermolecular hydrogen-bond stretching vibration in supercooled water, which was previously identified in the quenched normal mode spectrum of the TIP4P/2005 water model at low temperatures (70) (SI Appendix, Fig. S8) and may also be the origin of the small oscillatory behavior in the NGP time profile between 0.1 ps and 1 ps (Fig. 3B). We find that these oscillations in the MSD and NGP time profiles are absent in liquid water above the melting temperature (Fig. 2A) and hard disk fluids (SI Appendix, Fig. S1A) at any density and cannot be accurately represented by Eqs. 9 and 18, used for the model-based analysis of the MSD and NGP in the previous section.
The mean diffusion kernel and the diffusion kernel correlation extracted from the model-free analysis of the MSD and NGP transiently display oscillatory behaviors at times around 0.1 ps, which are more complicated than the behavior of their counterparts extracted using Eqs. 9 and 18 of the MSD and NGP (Fig. 3 C and D). However, at times shorter than 0.01 ps or longer than 0.4 ps, the model-free analysis yields essentially the same results as the model-based analysis. As shown in Fig. 3D, Inset, both the model-free and model-based methods yield essentially the same result for the long-time DKC or the TCF of the diffusion coefficient fluctuation, quantitatively explainable by the unbound-mode component, or the first term on the R.H.S. of Eq. 12, only. Consequently, both methods yield the same value for the whole-time integration of the DKC, or extrinsic disorder,
Microscopic Measurement of the Bound- and Unbound-Mode Components of Diffusion Kernel Correlation
To test the correctness of our results in the previous sections, we perform an alternative, microscopic measurement of the mean diffusion kernel and DKC using MD simulation and compare the results with those obtained in the previous sections. We then show that the bound-mode and unbound-mode transport dynamics, separately embodied in our transport model, clearly manifest, respectively, on the short-time and long-time dependence of the displacement distribution and the spatial volume spanned by the MD simulation trajectories for supercooled water at 193 K.
The mean diffusion kernel and DKC calculated from the MD simulation results of the two- and four-point TCF are found to be in excellent agreement with those extracted from the MSD and NGP. This is demonstrated for an example of supercooled water at 193 K in Figs. 3 and 4.
The short-time DKC, dominantly contributed from the bound-mode component, can be directly calculated using Eq. 6 and the direct MD simulation results of the VAFs; the BCF appearing in Eq. 6 can be calculated from its definition, Eq. 5. Note that the BCF can also be calculated from the MSD and NGP data, with use of the following relation:
We find that, at long times, the DKC is linearly proportional to the BCF (SI Appendix, Text S6):
An independent estimation of the long-time DKC can be made by using MD simulations to measure the diffusion coefficient fluctuation along each time trace and calculating the TCF of the diffusion coefficient (Methods). This is because the DKC becomes the diffusion coefficient at long times, as mentioned earlier. In Fig. 4C, for supercooled water at 193 K, we show that
An alternative estimation of the long-time DKC can be made from the time profile of the NGP. The long-time DKC is directly related to the NGP and its time derivatives as follows:
Bound-mode (unbound-mode) transport dynamics are reflected on the time dependence of the displacement distribution at short (long) times. For supercooled water at 193 K, the displacement distribution broadens rapidly before 1 ps but after this does not greatly change for several picoseconds (Fig. 4D). This bound-mode feature of transport dynamics also manifests itself on the time-dependent volume spanned by the MD simulation trajectories of a water molecule. As shown in Fig. 4E, the spatial volume spanned by the simulation trajectories rapidly increases with time before 1 ps, but afterward, this trajectory volume tends to saturate to a certain critical value over several picoseconds while the trajectory length continues increasing with time. In contrast, at long-time scales, the displacement distribution and the trajectory volume exhibit unbound-mode dynamics, as demonstrated in Fig. 4 F and G.
Quantitative Explanation of Fickian Yet Non-Gaussian Displacement Distribution
Disordered fluids exhibit non-Gaussian diffusion; that is, the displacement distribution is non-Gaussian even at long times where the MSD linearly increases with time (9, 12, 36⇓–38). The displacement distribution of disordered fluids starts as Gaussian with variance given by
To understand the time-dependent relaxation of the non-Gaussian displacement distribution in the Fickian diffusion regime, we need an explicit model of the diffusion coefficient fluctuation for the fluid system in question. In the literature, the diffusion coefficient is often modeled as
The first model assumes that the diffusion coefficient is given by
Quantitative explanation of displacement distributions for supercooled water and colloidal beads on lipid tubes. (A) Displacement distributions of a water molecule along a Cartesian coordinate at three different times,
In the second model, we model the diffusion coefficient as
Optimized values of adjustable parameters for colloidal bead diffusion on lipid tubes
The displacement distribution approaches a Gaussian distribution only after individual displacement trajectories become statistically equivalent. If individual displacement trajectories are statistically equivalent, the EB parameter proposed by He, Burov, Metzler, and Barkai (67) is linear in
Discussion
Main Findings.
We derived a transport equation, Eq. 2, describing stochastic thermal motion of various complex fluids, which yields exact analytic results, Eqs. 3a and 3b, that enable a unified, quantitative explanation of not only the MSD but also the NGP time profiles of various complex fluids (Fig. 2 and SI Appendix, Fig. S1). The central dynamic quantity governing transport dynamics of complex fluids is the environment-dependent diffusion kernel. The mean diffusion kernel (MDK) and DKC can be unambiguously extracted from the MSD and NGP time profiles (Fig. 3 and SI Appendix, Fig. S1). We also established an exact relationship of the MDK and DKC with the two-point and four-point VAFs (Eqs. 4–6), allowing for alternative, microscopic measurements of the MDK and DKC using MD simulation (Fig. 4 A–C). DKC is an ideal measure of mobility fluctuation of complex fluids exhibiting non-Fickian diffusion and is simply related to the NGP by Eq. 14, at long times (Fig. 4C).
We constructed a physical model of the environment-coupled diffusion kernel (Eqs. 10–12), composed of one unbound mode and multiple bound modes. This model provides a quantitative explanation of the MSD, NGP, and displacement distribution for various complex fluidic systems (Figs. 2 and 5 and SI Appendix, Fig. S1). Our model-based analysis of the frequency spectrum of the VAF suggests that the slight oscillation in supercooled water’s MSD originates from intramolecular hydrogen bond stretching motion. We introduced the notion of intrinsic disorder and extrinsic disorder for complex fluid systems in Eq. 8, which originate from a finite relaxation time of the mean diffusion kernel and the environment-coupled fluctuation of the diffusion kernel, respectively. We demonstrated a separate estimation of intrinsic and extrinsic disorder for supercooled water (Fig. 2E) and dense hard disk fluids (SI Appendix, Fig. S1E). Extrinsic disorder is more sensitive to temperature and density of complex fluids than intrinsic disorder; extrinsic disorder increases with inverse temperature and density, unless the complex fluids enter a solid-like phase.
Comparison with Previous Models.
Our model-based analysis of DKC is reminiscent of the memory kernel analyses by Berne, Boon, and Rice (77) and Douglas and Hubbard (78). Our random walk model is a generalization of the CTRW model, to account for environment-coupled fluctuation of transport dynamics. We refer to Shlesinger’s review (79) on the origins and applications of the CTRW model. Our model reduces to the CTRW when
Our transport equation, Eq. 2, encompasses the SD model, which accounts for extrinsic disorder but neglects intrinsic disorder. Eq. 2 reduces to the transport equation of the SD model when the diffusion kernel is replaced by the diffusion coefficient. The MSD and NGP of the SD model are given by
Potential Applications and Limitation.
Our transport equation can be extended for complex fluids under an external potential field or for nonergodic fluids such as glass (SI Appendix, Texts S2 and S4). It can be further generalized for dynamics of transport-coupled reactions in complex fluids (80⇓–82), which we leave for future research. Our MSD model in Eq. 9 is only approximate in the sense that it cannot capture weak oscillation in the MSD of supercooled water and the asymptotic long-time power-law relaxation of the two-point VAF of dense fluids (83⇓⇓–86). Improving our model to capture these phenomena is another future research topic.
In our transport equation, the diffusion kernel is independent of the absolute position of the tracer particle. However, by applying the projection operator technique (19, 87) to the Liouville equation, one can obtain a similar transport equation (SI Appendix, Text S16) but with the diffusion kernel dependent on the absolute position of the tracer particle, potentially important for a system with position-dependent transport dynamics. In most fluid systems, however, thermal motion is independent of the absolute position of the tracer particle.
For a more extensive discussion, see SI Appendix, Text S17, which we present at the request of an anonymous reviewer.
Outlook
The essential feature of our approach to transport dynamics of complex fluids is hidden environmental variables that represent the entire set of dynamic variables affecting transport dynamics of our tracer particles. By accounting for their effects without using an a priori explicit model, this approach enables the extraction of robust, quantitative information about the transport dynamics of complex fluid systems. This information can then be used to construct a more explicit model of the environment-coupled diffusion kernel,
Methods
Extraction of Diffusion Kernel Correlation from MSD and NGP.
Here, we present the procedure for extracting the DKC,
On the other hand, the NGP,
which can be rearranged with respect to the fourth moment,
The simulation results for the MSD and NGP are well represented by Eq. 9 with two bound modes (n = 2) and a linear combination of three or four Gaussian-shaped functions given by
respectively. We perform the best fits of Eqs. 9 and 18 to the simulation results for the MSD and NGP. By substituting the optimized results into Eq. 17, we obtain the analytic expression of
Generation of Time Traces of Fluctuating Diffusion Coefficient.
Along the ith particle trajectory,
Acknowledgments
We gratefully acknowledge Professors Mike Shlesinger, Eli Barkai, Ralf Metzler, YounJoon Jung, and Jae-Hyung Jeon for their helpful comments and Mr. Luke Bates for his careful reading of our manuscript. This work was supported by the Creative Research Initiative Project program (2015R1A3A2066497) and the National Research Foundation of Korea Grant (2015R1A2A1A15055664) funded by the Korean government. S.S. also acknowledges the Chung-Ang University Graduate Research Scholarship in 2016.
Footnotes
- ↵1To whom correspondence may be addressed. Email: jaeyoung{at}cau.ac.kr or jihyunkim{at}cau.ac.kr.
Author contributions: J.S. designed research; S.S., S.J.P., J.S.K., B.J.S., S.L., J.-H.K., and J.S. performed research; S.S., S.J.P., M.K., and J.-H.K. analyzed data; and S.S., J.S.K., B.J.S., S.L., J.-H.K., and J.S. wrote the paper.
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1900239116/-/DCSupplemental.
- Copyright © 2019 the Author(s). Published by PNAS.
This open access article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).
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- Transport Equation of Complex Fluids
- Analytic Expressions of the Moments
- Diffusion Kernel Correlation and Velocity Autocorrelation Functions
- Ergodicity and Long-Time Limit of Diffusion Kernel Correlation and Non-Gaussian Parameter
- Intrinsic and Extrinsic Disorder
- Model-Based Quantitative Analysis of the MSD and NGP
- Explicit Model for Diffusion Kernel
- Model-Free Quantitative Analysis of the MSD and NGP
- Microscopic Measurement of the Bound- and Unbound-Mode Components of Diffusion Kernel Correlation
- Quantitative Explanation of Fickian Yet Non-Gaussian Displacement Distribution
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