Skip to main content
  • Submit
  • About
    • Editorial Board
    • PNAS Staff
    • FAQ
    • Accessibility Statement
    • Rights and Permissions
    • Site Map
  • Contact
  • Journal Club
  • Subscribe
    • Subscription Rates
    • Subscriptions FAQ
    • Open Access
    • Recommend PNAS to Your Librarian
  • Log in
  • My Cart

Main menu

  • Home
  • Articles
    • Current
    • Special Feature Articles - Most Recent
    • Special Features
    • Colloquia
    • Collected Articles
    • PNAS Classics
    • List of Issues
  • Front Matter
  • News
    • For the Press
    • This Week In PNAS
    • PNAS in the News
  • Podcasts
  • Authors
    • Information for Authors
    • Editorial and Journal Policies
    • Submission Procedures
    • Fees and Licenses
  • Submit
  • About
    • Editorial Board
    • PNAS Staff
    • FAQ
    • Accessibility Statement
    • Rights and Permissions
    • Site Map
  • Contact
  • Journal Club
  • Subscribe
    • Subscription Rates
    • Subscriptions FAQ
    • Open Access
    • Recommend PNAS to Your Librarian

User menu

  • Log in
  • My Cart

Search

  • Advanced search
Home
Home

Advanced Search

  • Home
  • Articles
    • Current
    • Special Feature Articles - Most Recent
    • Special Features
    • Colloquia
    • Collected Articles
    • PNAS Classics
    • List of Issues
  • Front Matter
  • News
    • For the Press
    • This Week In PNAS
    • PNAS in the News
  • Podcasts
  • Authors
    • Information for Authors
    • Editorial and Journal Policies
    • Submission Procedures
    • Fees and Licenses

New Research In

Physical Sciences

Featured Portals

  • Physics
  • Chemistry
  • Sustainability Science

Articles by Topic

  • Applied Mathematics
  • Applied Physical Sciences
  • Astronomy
  • Computer Sciences
  • Earth, Atmospheric, and Planetary Sciences
  • Engineering
  • Environmental Sciences
  • Mathematics
  • Statistics

Social Sciences

Featured Portals

  • Anthropology
  • Sustainability Science

Articles by Topic

  • Economic Sciences
  • Environmental Sciences
  • Political Sciences
  • Psychological and Cognitive Sciences
  • Social Sciences

Biological Sciences

Featured Portals

  • Sustainability Science

Articles by Topic

  • Agricultural Sciences
  • Anthropology
  • Applied Biological Sciences
  • Biochemistry
  • Biophysics and Computational Biology
  • Cell Biology
  • Developmental Biology
  • Ecology
  • Environmental Sciences
  • Evolution
  • Genetics
  • Immunology and Inflammation
  • Medical Sciences
  • Microbiology
  • Neuroscience
  • Pharmacology
  • Physiology
  • Plant Biology
  • Population Biology
  • Psychological and Cognitive Sciences
  • Sustainability Science
  • Systems Biology
Research Article

Transport dynamics of complex fluids

View ORCID ProfileSanggeun Song, Seong Jun Park, Minjung Kim, Jun Soo Kim, Bong June Sung, Sangyoub Lee, Ji-Hyun Kim, and Jaeyoung Sung
PNAS June 25, 2019 116 (26) 12733-12742; first published June 7, 2019; https://doi.org/10.1073/pnas.1900239116
Sanggeun Song
aCreative Research Initiative Center for Chemical Dynamics in Living Cells, Chung-Ang University, 06974 Seoul, Republic of Korea;
bDepartment of Chemistry, Chung-Ang University, 06974 Seoul, Republic of Korea;
cNational Institute of Innovative Functional Imaging, Chung-Ang University, 06974 Seoul, Republic of Korea;
  • Find this author on Google Scholar
  • Find this author on PubMed
  • Search for this author on this site
  • ORCID record for Sanggeun Song
Seong Jun Park
aCreative Research Initiative Center for Chemical Dynamics in Living Cells, Chung-Ang University, 06974 Seoul, Republic of Korea;
bDepartment of Chemistry, Chung-Ang University, 06974 Seoul, Republic of Korea;
cNational Institute of Innovative Functional Imaging, Chung-Ang University, 06974 Seoul, Republic of Korea;
  • Find this author on Google Scholar
  • Find this author on PubMed
  • Search for this author on this site
Minjung Kim
dDepartment of Chemistry, College of Natural Sciences, Seoul National University, 08826 Seoul, Republic of Korea;
  • Find this author on Google Scholar
  • Find this author on PubMed
  • Search for this author on this site
Jun Soo Kim
eDepartment of Chemistry and Nanoscience, Ewha Womans University, 03760 Seoul, Republic of Korea;
  • Find this author on Google Scholar
  • Find this author on PubMed
  • Search for this author on this site
Bong June Sung
fDepartment of Chemistry, Sogang University, 04107 Seoul, Republic of Korea
  • Find this author on Google Scholar
  • Find this author on PubMed
  • Search for this author on this site
Sangyoub Lee
dDepartment of Chemistry, College of Natural Sciences, Seoul National University, 08826 Seoul, Republic of Korea;
  • Find this author on Google Scholar
  • Find this author on PubMed
  • Search for this author on this site
Ji-Hyun Kim
aCreative Research Initiative Center for Chemical Dynamics in Living Cells, Chung-Ang University, 06974 Seoul, Republic of Korea;
  • Find this author on Google Scholar
  • Find this author on PubMed
  • Search for this author on this site
  • For correspondence: jaeyoung@cau.ac.kr jihyunkim@cau.ac.kr
Jaeyoung Sung
aCreative Research Initiative Center for Chemical Dynamics in Living Cells, Chung-Ang University, 06974 Seoul, Republic of Korea;
bDepartment of Chemistry, Chung-Ang University, 06974 Seoul, Republic of Korea;
cNational Institute of Innovative Functional Imaging, Chung-Ang University, 06974 Seoul, Republic of Korea;
  • Find this author on Google Scholar
  • Find this author on PubMed
  • Search for this author on this site
  • For correspondence: jaeyoung@cau.ac.kr jihyunkim@cau.ac.kr
  1. 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)

  • Article
  • Figures & SI
  • Info & Metrics
  • PDF
Loading

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), ψ(t)∝t−(1+α) (0<α<1), successfully explains charge transport dynamics in amorphous semiconductors (29), which show a subdiffusive power-law MSD. These subdiffusive transport dynamics can be described by the fractional diffusion equation (30) or the fractional Fokker–Planck equation (31⇓–33) in the continuum limit. Mandelbrot and Van Ness’s (21) fractional Brownian motion (FBM) is another famous model of anomalous subdiffusive transport with a power-law MSD; however, FBM is a Gaussian subdiffusive process, while the CTRW with a power-law WTD is a non-Gaussian process. O’Shaughnessy and Procaccia (23) and Havlin and Ben-Avraham (24) investigate anomalous transport originating from self-similarity of transport media, considering random walks or diffusion in a fractal. Recently, Novikov et al. (34, 35) introduced a model of anomalous thermal motion dependent on mesoscopic structures of media and analyzed the long-time behavior of the diffusion coefficient by using a renormalization group solution.

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, ψΓ(t), coupled to arbitrary hidden environmental variables Γ (Fig. 1); Γ designates the entire set of dynamical variables that affect transport dynamics in disordered fluids. For this model, the joint probability, p(m,Γ,t), that a random walker is located at the mth site and the environmental state is at Γ at time t can be written as (49, 59)p(m,Γ,t)=∑N=0∞p(m|N)PN(Γ,t).[1]In Eq. 1, p(m|N) denotes the conditional probability that the random walker is located at the mth site, given that it has undergone N jumps, which is well known: p(m|N)=(N!/m+!m−!)2−N with m±=(N±m)/2 (N≥|m|), and p(m|N)=0 (N<|m|) (60). On the other hand, PN(Γ,t) denotes the joint probability that the total number of jumps made by the random walker is N and the environmental state is at Γ at time t. PN(Γ,t) is the crucial factor determining the time dependence of p(m,Γ,t). Using a generalized version of Sung and Silbey’s (61) master equation, which provides a formally exact description of the time evolution of PN(Γ,t), we derive the following transport equation from Eq. 1 in the continuum limit (SI Appendix, Text S1):p˙^(r,Γ,s)=D^Γ(s)∇2p^(r,Γ,s)+L(Γ)p^(r,Γ,s).[2]Here, p˙^(r,Γ,s) and p^(r,Γ,s) denote the Laplace transform of ∂p(r,Γ,t)/∂t and p(r,Γ,t), respectively; p(r,Γ,t) denotes the joint probability density that a particle is located at position r and the hidden environment is at state Γ at time t. This joint probability density satisfies the following normalization condition: ∫dr∫dΓp(r,Γ,t)=1. Throughout this work, f^(s) denotes the Laplace transform of f(t), i.e., ∫0∞dte−stf(t). In Eq. 2, D^Γ(s) designates the diffusion kernel that is determined by the environmental state-dependent sojourn time distribution, ψΓ(t) of our random walk model; i.e., D^Γ(s)= limε→0(ε2/2d)sψ^Γ(s)/[1−ψ^Γ(s)] =limε→0ε2κ^Γ(s)/2d with ε and d denoting the lattice constant and the spatial dimension, respectively. κ^Γ(s)(≡sψ^Γ(s)/[1−ψ^Γ(s)]) denotes the jump rate kernel of the random walker, which is dependent on lattice constant ε; for the continuum limit description, we assume limε→0ε2κ^Γ(s) exists. In Eq. 2, L(Γ) designates a mathematical operator describing the dynamics of the hidden environmental variables Γ.

Fig. 1.
  • Download figure
  • Open in new tab
  • Download powerpoint
Fig. 1.

Schematic representation of our random walk model with environmental state-dependent dynamics. In our model, the sojourn time distribution ψΓ(t) of a random walker is dependent on environmental state variables, Γ. The probabilistic dynamics of this random walker model can be described by Eq. 2 in the continuum limit.

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 L(Γ) at this point. A correct mathematical form of L(Γ) is dependent on the environment surrounding the system in question; when environmental state dynamics are a non-Markov process, L(Γ) may be dependent on Laplace variable s. As demonstrated in this work, quantitative information about transport dynamics coupled to hidden environmental variables can be extracted from simultaneous analysis of the MSD and NGP time profiles. This information can then be used to construct a more explicit model of transport dynamics of complex fluidic systems.

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, 〈|r(t)−r(0)|2〉(≡Δ2(t)) and 〈|r(t)−r(0)|4〉(≡Δ4(t)), of the displacement distribution:Δ^2(s)=2ds2〈D^Γ(s)〉,[3a]Δ^4(s)=(1+2d)2sΔ^2(s)2[1+sC^D(s)].[3b]In obtaining Eqs. 3a and 3b, we assume that the hidden environment is initially in a stationary state, such as the equilibrium state or the nonequilibrium steady state (see SI Appendix, Text S3 for details). The bracket notation, 〈⋯〉, designates the average over the stationary initial distribution of the environmental state. We can extend Eqs. 3a and 3b to the case where the initial state of the hidden environment is a nonstationary state, as is the case for glass, for which the analytic expressions of Δ^2(s) and Δ^4(s) are more complicated than Eqs. 3a and 3b (SI Appendix, Text S4). However, transport dynamics of various complex fluids can be quantitatively explained by Eqs. 3a and 3b, as demonstrated in this work. Hereafter, we focus on quantitative analysis of transport dynamics of complex fluidic systems with use of Eqs. 3a and 3b. We leave a quantitative explanation of transport dynamics in glass to future research.

The mean diffusion kernel, 〈DΓ(t)〉, in Eq. 3a is nothing but the two-point VAF; i.e., 〈DΓ(t)〉=〈v(t)⋅v(0)〉/d with v(t) being the velocity vector. This can be seen by comparing Eq. 3a and the Laplace transform of the well-known relation, Δ2(t)=2∫0tdτ2∫0τ2dτ1〈v(τ2−τ1)⋅v(0)〉(62), exact as long as 〈v(τ2)⋅v(τ1)〉 =〈v(τ2−τ1)⋅v(0)〉. Knowing this and utilizing the Tauberian theorem, we obtain lims→∞s〈D^Γ(s)〉=limt→0〈v(t)⋅v(0)〉/d=〈|v|2〉/d =kBT/M, with kBT and M denoting thermal energy and the mass of our tracer particle, respectively. This means that 〈D^Γ(s)〉 is proportional to the mean-square velocity in the large-s limit; i.e., 〈D^Γ(s)〉≅s−1〈|v|2〉/d (s→∞). On the other hand, in the small-s limit, the value of 〈D^Γ(s)〉 approaches ∫0∞dt〈v(t)·v(0)〉/d, which is simply the diffusion constant, D¯, according to the Green–Kubo relation (63). Substituting the small (large)-s limit asymptotic behavior of 〈D^Γ(s)〉 into Eq. 3a, we recover the well-known asymptotic behavior of the MSD: d(kBT/M)t2 at short times and 2dD¯t at long times.

While the second moment, Δ2(t), is dependent only on 〈DΓ(t)〉, the fourth moment, Δ4(t), is dependent on the environment-coupled fluctuation of the diffusion kernel, DΓ(t). In Eq. 3b, C^D(s) is the Laplace transform of the diffusion kernel correlation (DKC) or TCF of the diffusion kernel fluctuation (see SI Appendix, Eq. S3-12 for the precise definition). At long times, where the MSD is linear in time, the diffusion kernel becomes the diffusion coefficient; i.e., D^Γ(s)≅D^Γ(0)(≡DΓ) so that CD(t) can be identified as the TCF of the diffusion coefficient fluctuation; i.e., CD(t)≅〈δD(t)δD(0)〉/〈D〉2 =ηD2ϕD(t), where ηq2 and ϕq(t)(q∈{v2,D}) designate the relative variance, 〈δq2〉/〈q〉2, and the normalized TCF of q, 〈δq(t)δq(0)〉/〈δq2〉, respectively. At short times, on the other hand, CD(t) can be identified as the TCF of squared speed v2(t)(≡|v(t)|2); i.e., CD(t)≅dηv22ϕv2(t)[=d〈δv2(t)δv2(0)〉/〈v2〉2] (SI Appendix, Text S5). Given that the initial speed distribution obeys the Maxwell–Boltzmann distribution, we obtain 〈v4〉=(1+2/d)〈v2〉2, and the initial value of CD(t) can then only be given by limt→0CD(t)=d((〈v4〉−〈v2〉2))/〈v2〉2 =2. We find this is true for supercooled water and dense hard disk fluids (Fig. 2D and SI Appendix, Fig. S1D).

Fig. 2.
  • Download figure
  • Open in new tab
  • Download powerpoint
Fig. 2.

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, CD(t), extracted from the MSD and NGP data (Methods); dotted line, the mean-scaled TCF of squared speed of supercooled water at 193 K (SI Appendix, Fig. S11). In A, B, and D, the navy-blue triangles and the red squares represent the caging times, τc, and the NGP peak times, τng, respectively. In D, the yellow triangles represent the alpha relaxation times, τα (SI Appendix, Fig. S12). (E) Green circles, total disorder, limt→∞〈r2(t)〉α2(t)/σO2, with σO denoting an oxygen atom’s Lennard-Jones diameter, 3.1589 Å; yellow and cyan lines, intrinsic and extrinsic disorder (see text below Eq. 8); red circles, 4dC^D(0)/τD, where d, C^D(0), and τD(=σO2/D¯), respectively, denote the spatial dimension, the whole-time integration of CD(t), and the diffusion timescale. Tm and TW denote, respectively, the melting temperature (64) and the Widom line temperature (91, 92) (SI Appendix, Fig. S13).

In our theory, CD(t) is the essential dynamic quantity that characterizes environment-coupled mobility fluctuation. It serves as an ideal measure of mobility fluctuation for complex fluids exhibiting non-Fickian thermal motion, for which the TCF of the diffusion coefficient, the conventional measure of mobility fluctuation, is not well defined. As is shown below, there exists an exact relationship between CD(t) and the four-point and two-point VAFs, valid at all times and at any spatial dimension.

Using Eqs. 3a and 3b and the definition of the NGP, α2(t)[≡Δ4(t)/[(1+2/d)Δ2(t)2]−1], we can quantitatively explain the MSD and NGP of various complex fluids. From the MSD and NGP data, we can extract the time profiles of 〈DΓ(t)〉 and CD(t) either by assuming analytic functional forms for the MSD and NGP or without making any such assumption (Methods). In Fig. 2, we demonstrate our quantitative analysis of the molecular dynamics (MD) simulation results of the MSD and NGP for 4-point transferable intermolecular potential/2005 (TIP4P/2005) water (64), assuming specific analytic forms of the MSD and NGP but without assuming a particular model of the hidden environment or its influence on the diffusion kernel. As shown in this work, this information is useful in constructing an explicit model of transport dynamics of complex fluid systems; from this explicit model, we can predict or quantitatively understand the time dependence of the displacement distribution. Meanwhile, a model-free analysis of the MSD and NGP based on Eqs. 3a and 3b yields accurate and robust quantitative information about the time profiles of 〈DΓ(t)〉 and CD(t), but, on its own, is not physically interpretable (Fig. 3).

Fig. 3.
  • Download figure
  • Open in new tab
  • Download powerpoint
Fig. 3.

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 CD(t) in linear timescale. The difference between CD(t) and the unbound-mode contribution results from the presence of the bound modes, accounted for by the second term on the R.H.S. of Eq. 12 (SI Appendix, Fig. S14).

Diffusion Kernel Correlation and Velocity Autocorrelation Functions

The diffusion kernel correlation, CD(t), is closely related to the two- and four-point VAFs through the BCF. It is known that the NGP, α2(t), or the fourth cumulant, X4(t)[=(1+2/d)Δ2(t)2α2(t) =Δ4(t)−(1+2/d)Δ2(t)2], of displacement is related to the BCF, β(t), by (54⇓–56)X4(t)=4!d∫0tdt1(t−t1)β(t1),[4]where β(t) is defined byβ(t1)=∫0t1dt2∫0t2dt3[〈vα(0)vα(t1)vα(t2)vα(t3)〉−〈vα(0)vα(t1)〉〈vα(t2)vα(t3)〉−〈vα(0)vα(t2)〉〈vα(t1)vα(t3)〉−〈vα(0)vα(t3)〉〈vα(t1)vα(t2)〉].[5]In Eq. 5, vα indicates a Cartesian component of velocity vector, v, and 〈vα(0)vα(t1)vα(t2)vα(t3)〉 and 〈vα(0)vα(t)〉[=(kBT/M)ϕvα(t)] denote the four- and two-point VAFs, respectively. According to Wick’s theorem, β(t) vanishes when vα(t) is a Gaussian process. Taking the Laplace transform of Eq. 4, we obtain X^4(s)=Δ^4(s)−(1+2/d)Δ^22(s)=4!dβ^(s)/s2, where Δ^22(s) denotes the Laplace transform of Δ2(t)2. Substituting Eqs. 3a and 3b into the latter equation, we obtain an exact relation of the DKC to the BCF and the normalized VAF (see SI Appendix, Text S6 for the derivation):C^D(s)〈D^Γ(s)〉2=∫0∞dte−st[32+dβ(t)+(kBTM∫0tdt′ϕvα(t′))2 +kB2T2M2∫0tdt1∫0t1dt2[ϕvα(t)−ϕvα(t−t1)]ϕvα(t2)].[6]Using Eq. 6 along with Eq. 5 and 〈D^Γ(s)〉=(kBT/M)ϕ^vα(s), we can calculate the time profile of CD(t) from the four- and two-point VAFs, as demonstrated in Fig. 4B for supercooled water at 193 K.

Fig. 4.
  • Download figure
  • Open in new tab
  • Download powerpoint
Fig. 4.

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, β1(t) obtained from Eq. 5 and the four- and two-point VAFs obtained from simulation; blue circles, β2(t) obtained from β(t)=∂t2Χ4(t)/4!d and Χ4(t)=(1+2/d)Δ2(t)2α2(t). (Inset) The fourth cumulant, Χ4(t), of displacement. (B) Short-time diffusion kernel correlation, CD(t), obtained from (red circles) Eq. 6 and β1(t) and (blue circles) Eq. 6 and β2(t). (C) Long-time diffusion kernel correlation estimated by (squares) the mean-scaled TCF of diffusion coefficient fluctuation (Methods), (red dashed line) result of Eq. 14, and (Inset) time dependence of MSD scaled by 6t. The simulation results of the diffusion coefficient fluctuation are calculated using various bin times, ranging from 7 ns to 15 ns. In B and C, the black and yellow lines represent, respectively, the result of model-free theory and the contribution of the unbound mode to CD(t) shown in Fig. 3D. (D–G) Displacement distributions and representative time traces of three water molecules (D and E) at five different short times and (F and G) at three different long times. In E and G, the initial positions of all water molecules have been relocated to the center of the circle. In E, the dashed line represents a sphere centered at the initial position with radius Δ2(τc)1/2(≈0.6 Å).

Ergodicity and Long-Time Limit of Diffusion Kernel Correlation and Non-Gaussian Parameter

For fluidic systems showing terminal Fickian diffusion, CD(t) has the same long-time limit value as the NGP; i.e., CD(∞)=α2(∞), which can be shown by using Eqs. 3a and 3b and the definition of the NGP (SI Appendix, Text S7). This result indicates that the long-time limit NGP value, α2(∞), vanishes for ergodic fluid systems for which the TCF, CD(∞), of the diffusion coefficient vanishes in the long-time limit. However, for nonergodic systems with finite CD(∞), α2(∞) may not vanish, either. Therefore, α2(∞) can serve as an ergodicity measure for fluidic systems exhibiting long-time Fickian diffusion (SI Appendix, Fig. S2), which was noted by Odagaki (65) for the glass formation process. There exist transport systems with anomalous diffusion, Δ2(t)∝tv, and weak-ergodicity breaking. For such systems, α2(∞) deviates from CD(∞); even if CD(∞)=0, α2(∞) is finite and given by α2(∞)=vΓ(v)2/Γ(2v)−1 with Γ(z) denoting the Gamma function defined by Γ(z)=∫0∞dt tz−1e−t (SI Appendix, Text S7). This result was previously reported by Odagaki and Hiwatari (66) on the basis of the so-called coherent-medium approximation, which corresponds to a vanishingly small DKC; i.e., CD(t)=0. Finally, for nonergodic systems exhibiting anomalous diffusion, Δ2(t)∝tv, at long times, we find that the relationship between CD(∞) and α2(∞) deviates from the result for the weak-ergodicity breaking system; instead, α2(∞) is given by α2(∞)=vΓ(v)2[1+CD(∞)]/Γ(2v)−1 (SI Appendix, Text S7). These results suggest that the finite value of α2(∞) can serve an alternative measure of nonergodicity, which is constant in time unlike the ergodicity-breaking (EB) parameter proposed by He, Burov, Metzler, and Barkai (67).

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:Δ2(t)≅2dD¯t+Δc (t→∞)[7a]α2(t)≅2C^D(0)+Δc/dD¯t (t→∞).[7b]Eq. 7a can be obtained by substituting the Maclaurin series of 〈D^Γ(s)〉, 〈D^Γ(s)〉=(ε2/2d)〈κ^Γ(s)〉=(ε2/2d) × [〈κ^Γ(s)〉+〈κ^′Γ(s)〉s+⋯], into Eq. 3a and by taking the inverse Laplace transform of the resulting equation. We present the derivation of Eqs. 7a and 7b in SI Appendix, Text S7. In Eq. 7a, Δc is given by Δc=2∫0∞dt〈DΓ(t)〉t =2∫0∞dt〈v(t)⋅v(0)〉t and vanishes only when the VAF decays infinitely fast or only when velocity is white noise. In our random walk model, Δc emerges whenever the waiting-time distribution is a nonexponential function (SI Appendix, Text S8). On the other hand, in Eq. 7b, C^D(0)[=∫0∞dtCD(t)] emerges whenever the diffusion kernel or the waiting-time distribution is coupled to environmental variables.

From Eqs. 7a and 7b, we obtain the long-time limit value of the product of the MSD and NGP aslimt→∞〈r2(t)〉α2(t)=2[Δc+2dD¯C^D(0)].[8]We define the dimensionless disorder strength of complex fluids as limt→∞〈r2(t)〉α2(t)/σ2, with σ being the effective diameter of a tracer particle. Eq. 8 tells us that disorder strength has two different components, 2Δc/σ2 and 4dD¯C^D(0)/σ2, which originate from a finite relaxation time of the mean diffusion kernel and from environment-coupled fluctuation of the diffusion kernel, respectively. We designate the latter term extrinsic disorder, which is quite sensitive to temperature and density of the environment, as demonstrated in Fig. 2E and SI Appendix, Fig. S1E. On the other hand, we designate 2Δc/σ2 intrinsic disorder, because this term persists even when environment-coupled fluctuation in transport dynamics is negligible.

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 4dD¯C^D(0)/σ2 or, more simply, by subtracting intrinsic disorder from total disorder strength (Fig. 2E); i.e., 4dD¯C^D(0)/σ2 =[limt→∞〈r2(t)〉α2(t)−2Δc]/σ2.

Disorder strength, defined in Eq. 8, is directly related to the whole-time integration of the BCF; that is, limt→∞〈r2(t)〉α2(t)/2 =Δc+2dD¯C^D(0) =6d2+dβ^(0)/D¯. This follows from the small-s limit of X^4(s)=Δ^4(s)−(1+2/d)Δ^22(s) =4!dβ^(s)/s2, obtained from Eq. 4, and from Eqs. 3a and 3b (SI Appendix, Text S6).

Model-Based Quantitative Analysis of the MSD and NGP

Intrinsic disorder causes the MSD to deviate from 2dD¯t, the prediction of the simple diffusion equation. We find that the following formula provides an excellent approximate description of the entire time range of the MSD for various disordered fluids (SI Appendix, Text S9):Δ2(t)=2dkBTMγ02c0(γ0t−1+e−γ0t)+2dkBTM∑i=1nciω0,i2[1−e−γit(cosh⁡ωit+γiωisinh⁡ωit)].[9]This equation represents the MSD of a bead in a Gaussian polymer, but also quantitatively explains the MSD of liquid water and dense hard disk fluids (Fig. 2A and SI Appendix, Fig. S1A).

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, ci and γi designate the weight coefficient and relaxation rate of the ith mode (0≤i≤n). The weight coefficients are normalized by ∑i=0nci=1. ω0,i is the natural frequency of the ith bound mode and is related to ωi as ωi=γi2−ω0,i2.

At all times, temperatures, and densities investigated, Eq. 9 with only two bound modes (n=2) already provides a good quantitative explanation of the simulation results for the anomalous MSD of supercooled water and hard disk fluids (Fig. 2A and SI Appendix, Fig. S1A) and the experimental results for colloidal beads moving along lipid tubes (SI Appendix, Fig. S3) (36). According to Eq. 3a, the analytic expression of the mean diffusion kernel yielding the MSD given in Eq. 9 can be obtained by〈DΓ(t)〉=kBTMc0e−γ0t+kBTM∑i=1ncie−γit[cosh(ωit)−γiωisinh(ωit)].[10]Fig. 2C shows the mean diffusion kernel, or the VAF, calculated from Eq. 10 with parameter values optimized against MSD data from MD simulation shown in Fig. 2A for supercooled water.

The NGP is dependent not only on the mean transport dynamics, 〈D^Γ(s)〉, but also on fluctuation of transport dynamics, CD(t). For simple diffusion, we have 〈D^Γ(s)〉=D¯ and CD(t)=0, and Eqs. 3a and 3b yield Δ2(t)=2dD¯t and Δ4(t)=(1+2/d)Δ2(t)2 so that the NGP vanishes. However, whenever 〈DΓ(t)〉 is not constant and/or CD(t)≠0, the NGP does not vanish. We find that, for disordered fluid systems investigated in this work, the time profiles of the NGP cannot be quantitatively understood when we neglect fluctuation in the diffusion kernel or when we assume CD(t)=0 (SI Appendix, Fig. S4).

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, α2(t)∝t2 at short times (SI Appendix, Text S10 and Eq. S10-9), but decreases with time, α2(t)∝t−1, at long times following Eq. 7b. As shown in Fig. 2A, it is only after the NGP peak time that Fickian diffusion emerges. These properties of the NGP are not specific to supercooled water but common across various disordered fluids (9, 12).

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, α2(τng), serves as a measure of the relative variance of the diffusion coefficient for supercooled water. From the displacement distribution at the NGP peak time, we can extract the distribution of the diffusion coefficient using the method proposed in ref. 37. We find the relative variance, ηD2, of the extracted diffusion coefficient distribution has the same value as the NGP peak height (SI Appendix, Fig. S5D). This is not a coincidence. We find the NGP peak height has the same value as the relative variance of the diffusion coefficient at the Fickian diffusion onset time or the NGP peak time, τng (SI Appendix, Text S11). Both the NGP peak height and the NGP peak time increase with inverse temperature and density (Fig. 2B and SI Appendix, Fig. S1B).

Explicit Model for Diffusion Kernel

In the previous section, we demonstrated that the time profile of CD(t) can be extracted from the MSD and the NGP based on Eqs. 3a and 3b using Eq. 9 for the MSD. To achieve a physical understanding of the time profile of CD(t), we construct an explicit model of the environment-coupled diffusion kernel. Let us first consider the Laplace transform of Eq. 10, 〈D^Γ(s)〉=c0f^0(s)+∑i=1ncif^i(s), where f^0(s) and f^i(s) denote the diffusion kernels associated with the unbound-mode dynamics and the ith bound-mode dynamics, given by f^0(s)=(kBT/M)(s+γ0)−1 and f^i(s)=(kBT/M)s[(s+γi)2−ωi2]−1, respectively. We can extend this equation by assuming the weight coefficients {c0,c1,⋯,cn} are dependent on environmental state variables Γ, obtaining the following model of the diffusion kernel:D^Γ(s)=c0(Γ)f^0(s)+∑i=1nci(Γ)f^i(s).[11]From this model, we obtain the analytic expression for C^D(s) (SI Appendix, Text S12),C^D(s)=〈D^Γ(s)〉−2[γ02〈δD2〉ϕ^D(s)(s+γ0)2+∑i=0n∑j=0n′C^ij(s)f^i(s)f^j(s)],[12]where the prime notation signifies that the sum excludes the term with i=j=0, and Cij(t) denotes the time correlation between weight coefficients; i.e., Cij(t)=〈δci(t)δcj(0)〉. Noting that lims→0f^0(s)=kBT/Mγ0 and lims→0f^i>0(s)=0, we obtain D^Γ(s)≅c0(Γ)(kBT/Mγ0)(≡DΓ) from Eq. 11 in the small-s regime, s≪γi. Therefore, we can relate the weight coefficient TCF, 〈δc0(t)δc0(0)〉, of the unbound mode to the TCF of the diffusion coefficient fluctuation by 〈δc0(t)δc0(0)〉 =〈δD(t)δD(0)〉(Mγ0/kBT)2 at times longer than any element of {γi−1}. We find that the largest element of {γi−1} is γ0−1, whose value has order of 1 ps for the unbound mode for supercooled water (Table 1). At times longer than the NGP peak time, τng, which is greater than the largest velocity relaxation time, γ0−1 (Fig. 2 B and C), the bound-mode terms are negligible compared with unbound-mode term, so that the first term on the R.H.S. of Eq. 12 contributes the most to the relaxation of diffusion kernel fluctuation, leaving us with CD(t)≅ϕD(t)ηD2 (t>τng). This result indicates that the DKC becomes the TCF of the diffusion coefficient at long times. Using this result and recalling that ηD2≅α2(τng) (SI Appendix, Fig. S5D), we can then extract ϕD(t) from the time profile of CD(t)/α2(τng) at times longer than the NGP peak time. The long-time tail of CD(t) or ϕD(t) extracted from the MSD and NGP can be explained by an explicit model of the diffusion coefficient fluctuation described later in this work.

View this table:
  • View inline
  • View popup
Table 1.

Optimized values of adjustable parameters for supercooled water at 193 K

We note here that the whole-time integration of the diffusion kernel, CD(t), the determining factor of extrinsic disorder, is mostly contributed from the unbound-mode component (SI Appendix, Fig. S6). The bound-mode contribution to CD(t) has a comparable magnitude to the unbound-mode contribution, but has a negligibly smaller relaxation timescale than the unbound-mode contribution; consequently, the unbound-mode component makes the dominant contribution to C^D(0)=∫0∞dtCD(t). The major contributor to CD(t) is the unbound mode at long times but the bound modes at short times. For example, for supercooled water at 193 K, the bound-mode components of CD(t) are dominant at times shorter than 30 ps, and the unbound-mode component is dominant at times longer than 5 ns (Fig. 3D).

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, 4dD¯C^D(0)/σ2(=[limt→∞〈r2(t)〉α2(t)−2Δc]/σ2), and hence for intrinsic disorder as well, because total disorder can only be the sum of intrinsic and 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: β(t)=∂t2X4(t)/4!d and X4(t)=(1+2/d)Δ2(t)2α2(t). These two methods produce similar, noisy BCF time profiles as shown in Fig. 4A for supercooled water at 193 K. When substituted into Eq. 6, these two BCF time profiles yield essentially the same results for the DKC, which is also in perfect agreement with the DKC extracted from our analysis of the MSD and NGP in the previous section (Fig. 4D).

We find that, at long times, the DKC is linearly proportional to the BCF (SI Appendix, Text S6):CD(t)≅32+dβ(t)/D¯2.[13]Recalling that the DKC becomes the TCF of the diffusion coefficient fluctuation, CD(t)≅ϕD(t)ηD2 =〈δD(t)δD(0)〉/D¯2 at long times, we obtain the long-time approximation of Eq. 13 as 〈δD(t)δD(0)〉≅3β(t)/(d+2). This result tells us that, for a one-dimensional system, the BCF is the same as the diffusion coefficient fluctuation, which was previously recognized by Nieuwenhuizen and Ernst (54) and others (71, 72) for a one-dimensional system of independent charged particles hopping on a lattice with static disorder. Our result here indicates that the long-time BCF has the same time profile as the long-time DKC multiplied by (d+2)/3, which is shown in Fig. 3D, Inset. It is not easy to calculate the long-time profile of the BCF directly from its definition, Eq. 5, because of the large computational expense involved in accurately estimating the multipoint VAFs from MD simulation and the 2D integral appearing in Eq. 5.

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 ηD2ϕD(t) calculated from direct MD simulation is actually in good agreement with the long-time profile of CD(t) extracted from the MSD and NGP data. The long-time DKC is dominantly contributed from the unbound-mode component, or the first term on the R.H.S. of Eq. 12, as shown in Fig. 4C.

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:CD(t)≅12∂2∂t2[t2α2(t)]=α2(t)+2tα˙2(t)+2−1t2α¨(t).[14]This equation can be obtained by substituting the long-time expression of the fourth cumulant, X4(t)≅(1+2/d)(2dD¯t)2α2(t), into the well-known equation, β(t)=∂t2X4(t)/4!d, and then substituting the result into Eq. 13. Eq. 14 tells us that the NGP carries the complete information about the long-time relaxation of mobility fluctuation of complex fluids. As shown in Fig. 4C for supercooled water at 193 K, the long-time DKC estimated by Eq. 14 quantitatively agrees with the long-time DKC obtained from three other methods, namely, extraction from the MSD and NGP data, using Eq. 6 and MD simulation results of the VAF, and MD simulation of the diffusion coefficient fluctuation and calculation of its TCF.

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 r(Δt)−r(0)≅v(0)Δt at short times but deviates at any finite time. At the NGP peak time, where the deviation from Gaussian is greatest, the displacement distribution often appears similar to a Laplace distribution with an exponential tail (50). It is at this time that the non-Gaussian displacement distribution begins its relaxation to Gaussian and the MSD becomes linear in time. This phenomenon is widely observed across various disordered fluid systems (7, 9, 12), but has yet to be quantitatively explained.

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 D=A⁡exp(−βE), where A, β, and E are the entropic factor, inverse thermal energy, and activation energy, respectively (73). This model can be generalized by assuming that the entropic factor and activation energy are stochastic variables dependent on hidden environmental variables. In this work, we make this generalization and consider two exactly solvable models.

The first model assumes that the diffusion coefficient is given by DΓ=AΓ⁡exp(−βEΓ), where the fluctuation of EΓ around its mean value, 〈EΓ〉, is given by EΓ(t)−〈E〉=∑kbkΓk(t), where {bk} and AΓ are constants and {Γk} are stationary Gaussian Markov processes, also known as Ornstein–Uhlenbeck (OU) processes, satisfying 〈Γk(t)Γl(0)〉=δkl⁡exp(−λkt) (SI Appendix, Text S13) (74). For this model, exact analytic expressions of 〈D〉, ηD2, and ϕD(t)ηD2 can be obtained (Table 1 legend and SI Appendix, Eq. S13-3), where the adjustable parameters are optimized against the diffusion coefficient value, the NGP peak value, and the long-time DKC data (SI Appendix, Text S13). The optimized parameter values are presented in Table 1. Using the first model with optimized parameter values, we can now predict the time-dependent relaxation of the non-Gaussian displacement distribution in the Fickian diffusion regime (SI Appendix, Text S13). The prediction of this model is in excellent agreement with the MD simulation results for the displacement distribution of supercooled water, as shown in Fig. 5A.

Fig. 5.
  • Download figure
  • Open in new tab
  • Download powerpoint
Fig. 5.

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, τα(≅2.9ns), 5τα(≅14.4ns), and τEB(≅36.4ns). Circles, simulation results for TIP4P/2005 water at 193 K; lines, theoretical predictions of our first model. (B) Scaled displacement distribution of colloidal beads with diameter σ along a lipid tube at various times. Circles, experimental results reported in ref. 36; lines, theoretical results of our second model (SI Appendix, Texts S12 and S14).

In the second model, we model the diffusion coefficient as DΓ(t)=AΓ(t)exp(−βEΓ(t)) ≅〈D〉∑kakΓk2(t), where {ak} and {Γk(t)} are constants and OU processes, respectively. This is a generalization of the model in ref. 47 and yields an analytic expression of the displacement distribution (Table 2 legend and SI Appendix, Eq. S14-3). We find this expression provides an excellent quantitative explanation of the experimentally measured displacement distribution of colloidal beads diffusing on lipid tubes reported in ref. 36 (Fig. 5B). The optimized parameters of the second model are presented in Table 2. Eq. 12 with optimized parameter values allows us to calculate the time profiles of the DKC for the colloidal bead system (Table 2 legend and SI Appendix, Fig. S3).

View this table:
  • View inline
  • View popup
Table 2.

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 t/tmax, where t and tmax denote the time lag, or the interval over which the time-averaged MSD is calculated, and the maximum trajectory length, respectively (75). Otherwise, the EB parameter deviates from its linear dependence on t/tmax. At temperatures lower than 230 K, the EB parameter of supercooled water shows an anomalous power-law dependence on t/tmax at short times but resumes normal linear dependence on t/tmax at times longer than the characteristic time τEB (SI Appendix, Fig. S9). Deviation of the displacement distribution from Gaussian becomes negligible only at times much longer than τEB, as demonstrated in SI Appendix, Fig. S9B for supercooled water. On the other hand, CD(t) is negligibly small at the characteristic time τEB (SI Appendix, Fig. S9B). This is due to the fact that the long-time relaxation of the NGP is contributed not only from extrinsic disorder leading to the trajectory-to-trajectory variation in the transport dynamics, but also from intrinsic disorder, whose effects persist even for homogeneous systems with statistically equivalent displacement trajectories (Eq. 7b). This analysis shows that the long-time tail of CD(t) better characterizes the relaxation of diffusivity fluctuation than the NGP (76).

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 CD(t)=0; both models yield the same MSD but different NGPs (SI Appendix, Figs. S2E and S4). For the CTRW model, the NGP is given by α2(t)≅−2/3 (t→0) (SI Appendix, Eq. S10-4); (Δc/dD¯)t−1 (t→∞). The MSD of a random walker is dependent on the initial condition; non-Fickian diffusion of complex fluids cannot be modeled by a random walker model with a stationary initial condition (SI Appendix, Fig. S10 and Text S15).

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 Δ2(t)=2dD¯t and α2(t)≅ηD2 (t→0); 2ηD2ϕ^D(0)t−1 (t→∞) (SI Appendix, Text S11), which cannot describe complex fluids with a non-Fickian MSD and initially vanishing NGP.

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, D^Γ(s), which has proved to be useful in quantitative interpretation and prediction of the MSD, NGP, and displacement distribution of various complex fluid systems. In achieving quantitative understanding of complex systems, this type of approach is advantageous over the conventional approach that relies on fully explicit models of the system, the environment, and their interactions. This is because, for a system interacting with a complex environment, it is difficult to construct a model that is both accurate and explicit from the outset, due to lack of information. Our approach is applicable and extendable to quantitative investigation of various other complex systems in natural science (88, 89).

Methods

Extraction of Diffusion Kernel Correlation from MSD and NGP.

Here, we present the procedure for extracting the DKC, CD(t), from the MSD and NGP obtained by computer simulation, where CD(t) is defined by SI Appendix, Eq. S3-12 in the Laplace domain. From SI Appendix, Eq. S3-12, we can represent C^D(s) in terms of the first two nonvanishing moments as follows:C^D(s)=Δ^4(s)(1+2d)2s2Δ^2(s)2−1s.[15]

On the other hand, the NGP, α2(t), is defined byα2(t)=Δ4(t)(1+2d)Δ2(t)2−1,[16]

which can be rearranged with respect to the fourth moment, Δ4(t), asΔ4(t)=(1+2d)Δ2(t)2[1+α2(t)].[17]

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 byα2(t)≅∑i=13or4ai⁡exp[−(log10t−bi)2/ci],[18]

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 Δ4(t) as a function of time. Taking the Laplace transforms of the best fitted Δ2(t) and Δ4(t), and substituting the results into Eq. 15, we obtain the Laplace transform of CD(t) for a given set of MSD and NGP data. Here, we can directly use the model-free results for the MSD and NGP obtained from the simulation, instead of using the model-based fits. To obtain the value of CD(t) at a given time, t, we perform the numerical Laplace inversion of Eq. 15 using the Stehfest algorithm (90).

Generation of Time Traces of Fluctuating Diffusion Coefficient.

Along the ith particle trajectory, N+1 displacements, ri(te+tj)−ri(tj), with the same elapsed time, te, but different starting times, {tj|k≤j≤N+k}, are selected. Here, {tj} are successive time sequences with a constant spacing, 2 ps. The elapsed time, te, must be long enough to capture Fickian diffusion. As shown in Fig. 4C, Inset, the MSD of a TIP4P/2005 water molecule at 193 K becomes fully linear in time beyond roughly 10 ns. From the selected displacements, the time-local MSD, Δ2(i)(te;tk) =N−1∑j=kN+k[ri(tj+te)−ri(tj)]2, conditioned on time tk for the ith particle trajectory is calculated. Δ2(i)(te;tk)/6te is then assigned as a diffusion coefficient, D(i)(tk), at time tk (54, 71, 72). The resulting mean-scaled TCF, 〈δD(t)δD(0)〉/〈D〉2, obtained with N = 20 and several elapsed times around 10 ns are well superimposed on each other (Fig. 4C). Here, the elapsed time, te, must be smaller than t as noted in ref. 71. It is also verified that the mean-scaled TCF obtained with N = 10 or 30 is essentially the same as the result with N = 20. When te=10 ns, the values of mean, 〈D〉, and relative variance, ηD2, for the diffusion coefficient are estimated to be 3.2 × 10−6 m2⋅s−1 and 1.2, respectively, consistent with D¯=limt→∞Δ2(t)/6t= 3.1 × 10−6 m2⋅s−1 and α2(τng)=1.3.

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).

View Abstract

References

  1. ↵
    1. A. Einstein
    , On the motion of small particles suspended in liquids at rest required by the molecular-kinetic theory of heat. Ann. Phys. (Leipzig) 17, 549–560 (1905).
    OpenUrl
  2. ↵
    1. C. Di Rienzo,
    2. V. Piazza,
    3. E. Gratton,
    4. F. Beltram,
    5. F. Cardarelli
    , Probing short-range protein Brownian motion in the cytoplasm of living cells. Nat. Commun. 5, 5891 (2014).
    OpenUrl
  3. ↵
    1. Y. Golan,
    2. E. Sherman
    , Resolving mixed mechanisms of protein subdiffusion at the T cell plasma membrane. Nat. Commun. 8, 15851 (2017).
    OpenUrl
  4. ↵
    1. M. Doi,
    2. S. Edwards
    , Dynamics of concentrated polymer systems. Part 1.—Brownian motion in the equilibrium state. J. Chem. Soc. Faraday Trans. 74, 1789–1801 (1978).
    OpenUrl
  5. ↵
    1. F. Sciortino,
    2. P. Gallo,
    3. P. Tartaglia,
    4. S. Chen
    , Supercooled water and the kinetic glass transition. Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 54, 6331–6343 (1996).
    OpenUrlPubMed
  6. ↵
    1. S. D. Overduin,
    2. G. N. Patey
    , An analysis of fluctuations in supercooled TIP4P/2005 water. J. Chem. Phys. 138, 184502 (2013).
    OpenUrl
  7. ↵
    1. M. G. Del Pópolo,
    2. G. A. Voth
    , On the structure and dynamics of ionic liquids. J. Phys. Chem. B 108, 1744–1752 (2004).
    OpenUrlCrossRef
  8. ↵
    1. Z. Hu,
    2. C. J. Margulis
    , Heterogeneity in a room-temperature ionic liquid: Persistent local environments and the red-edge effect. Proc. Natl. Acad. Sci. U.S.A. 103, 831–836 (2006).
    OpenUrlAbstract/FREE Full Text
  9. ↵
    1. J. Kim,
    2. C. Kim,
    3. B. J. Sung
    , Simulation study of seemingly Fickian but heterogeneous dynamics of two dimensional colloids. Phys. Rev. Lett. 110, 047801 (2013).
    OpenUrl
  10. ↵
    1. L. Larini,
    2. A. Ottochian,
    3. C. De Michele,
    4. D. Leporini
    , Universal scaling between structural relaxation and vibrational dynamics in glass-forming liquids and polymers. Nat. Phys. 4, 42 (2008).
    OpenUrlCrossRef
  11. ↵
    1. P. Charbonneau,
    2. Y. Jin,
    3. G. Parisi,
    4. F. Zamponi
    , Hopping and the Stokes-Einstein relation breakdown in simple glass formers. Proc. Natl. Acad. Sci. U.S.A. 111, 15025–15030 (2014).
    OpenUrlAbstract/FREE Full Text
  12. ↵
    1. B. van der Meer,
    2. W. Qi,
    3. J. Sprakel,
    4. L. Filion,
    5. M. Dijkstra
    , Dynamical heterogeneities and defects in two-dimensional soft colloidal crystals. Soft Matter 11, 9385–9392 (2015).
    OpenUrl
  13. ↵
    1. N. Wax
    , Selected Papers on Noise and Stochastic Processes (Courier Dover Publications, 1954).
  14. ↵
    1. Mv. Smoluchowski
    , Versuch einer mathematischen theorie der koagulationskinetik kolloider lösungen. Z. Phys. Chem. 92, 129–168 (1918).
    OpenUrl
  15. ↵
    1. L. Onsager
    , Initial recombination of ions. Phys. Rev. 54, 554 (1938).
    OpenUrlCrossRef
  16. ↵
    1. P. Debye
    , Reaction rates in ionic solutions. Trans. Electrochem. Soc. 82, 265–272 (1942).
    OpenUrlCrossRef
  17. ↵
    1. P. E. Rouse Jr
    , A theory of the linear viscoelastic properties of dilute solutions of coiling polymers. J. Chem. Phys. 21, 1272–1280 (1953).
    OpenUrlCrossRef
  18. ↵
    1. B. H. Zimm
    , Dynamics of polymer molecules in dilute solution: Viscoelasticity, flow birefringence and dielectric loss. J. Chem. Phys. 24, 269–278 (1956).
    OpenUrlCrossRef
  19. ↵
    1. H. Mori
    , Transport, collective motion, and Brownian motion. Prog. Theor. Phys. 33, 423–455 (1965).
    OpenUrlCrossRef
  20. ↵
    1. E. W. Montroll,
    2. G. H. Weiss
    , Random walks on lattices. II. J. Math. Phys. 6, 167–181 (1965).
    OpenUrlCrossRef
  21. ↵
    1. B. B. Mandelbrot,
    2. J. W. Van Ness
    , Fractional Brownian motions, fractional noises and applications. SIAM Rev. 10, 422–437 (1968).
    OpenUrl
  22. ↵
    1. K. Kawasaki
    , Kinetic equations and time correlation functions of critical fluctuations. Ann. Phys. 61, 1–56 (1970).
    OpenUrl
  23. ↵
    1. B. O’Shaughnessy,
    2. I. Procaccia
    , Analytical solutions for diffusion on fractal objects. Phys. Rev. Lett. 54, 455–458 (1985).
    OpenUrlCrossRefPubMed
  24. ↵
    1. S. Havlin,
    2. D. Ben-Avraham
    , Diffusion in disordered media. Adv. Phys. 36, 695–798 (1987).
    OpenUrlCrossRef
  25. ↵
    1. J. Klafter,
    2. A. Blumen,
    3. M. F. Shlesinger
    , Stochastic pathway to anomalous diffusion. Phys. Rev. A Gen. Phys. 35, 3081–3085 (1987).
    OpenUrl
  26. ↵
    1. M. F. Shlesinger,
    2. G. M. Zaslavsky,
    3. J. Klafter
    , Strange kinetics. Nature 363, 31–37 (1993).
    OpenUrlCrossRef
  27. ↵
    1. B. Berkowitz,
    2. H. Scher
    , Theory of anomalous chemical transport in random fracture networks. Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 57, 5858 (1998).
    OpenUrl
  28. ↵
    1. J. Klafter,
    2. R. Silbey
    , Derivation of the continuous-time random-walk equation. Phys. Rev. Lett. 44, 55 (1980).
    OpenUrlCrossRef
  29. ↵
    1. H. Scher,
    2. E. W. Montroll
    , Anomalous transit-time dispersion in amorphous solids. Phys. Rev. B 12, 2455 (1975).
    OpenUrl
  30. ↵
    1. W. Wyss
    , The fractional diffusion equation. J. Math. Phys. 27, 2782–2785 (1986).
    OpenUrlCrossRef
  31. ↵
    1. R. Metzler,
    2. E. Barkai,
    3. J. Klafter
    , Anomalous diffusion and relaxation close to thermal equilibrium: A fractional Fokker-Planck equation approach. Phys. Rev. Lett. 82, 3563 (1999).
    OpenUrlCrossRef
  32. ↵
    1. E. Barkai,
    2. R. Metzler,
    3. J. Klafter
    , From continuous time random walks to the fractional Fokker-Planck equation. Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 61, 132–138 (2000).
    OpenUrlPubMed
  33. ↵
    1. E. Barkai
    , CTRW pathways to the fractional diffusion equation. Chem. Phys. 284, 13–27 (2002).
    OpenUrl
  34. ↵
    1. D. S. Novikov,
    2. E. Fieremans,
    3. J. H. Jensen,
    4. J. A. Helpern
    , Random walk with barriers. Nat. Phys. 7, 508–514 (2011).
    OpenUrlCrossRefPubMed
  35. ↵
    1. D. S. Novikov,
    2. J. H. Jensen,
    3. J. A. Helpern,
    4. E. Fieremans
    , Revealing mesoscopic structural universality with diffusion. Proc. Natl. Acad. Sci. U.S.A. 111, 5088–5093 (2014).
    OpenUrlAbstract/FREE Full Text
  36. ↵
    1. B. Wang,
    2. S. M. Anthony,
    3. S. C. Bae,
    4. S. Granick
    , Anomalous yet Brownian. Proc. Natl. Acad. Sci. U.S.A. 106, 15160–15164 (2009).
    OpenUrlAbstract/FREE Full Text
  37. ↵
    1. B. Wang,
    2. J. Kuo,
    3. S. C. Bae,
    4. S. Granick
    , When Brownian diffusion is not Gaussian. Nat. Mater. 11, 481–485 (2012).
    OpenUrlCrossRefPubMed
  38. ↵
    1. J. Guan,
    2. B. Wang,
    3. S. Granick
    , Even hard-sphere colloidal suspensions display Fickian yet non-Gaussian diffusion. ACS Nano 8, 3331–3336 (2014).
    OpenUrl
  39. ↵
    1. S.-W. Park,
    2. S. Kim,
    3. Y. Jung
    , Time scale of dynamic heterogeneity in model ionic liquids and its relation to static length scale and charge distribution. Phys. Chem. Chem. Phys. 17, 29281–29292 (2015).
    OpenUrl
  40. ↵
    1. J.-H. Jeon,
    2. M. Javanainen,
    3. H. Martinez-Seara,
    4. R. Metzler,
    5. I. Vattulainen
    , Protein crowding in lipid bilayers gives rise to non-Gaussian anomalous lateral diffusion of phospholipids and proteins. Phys. Rev. X 6, 021006 (2016).
    OpenUrl
  41. ↵
    1. S. Kim,
    2. S.-W. Park,
    3. Y. Jung
    , Heterogeneous dynamics and its length scale in simple ionic liquid models: A computational study. Phys. Chem. Chem. Phys. 18, 6486–6497 (2016).
    OpenUrl
  42. ↵
    1. N. Tyagi,
    2. B. J. Cherayil
    , Non-Gaussian Brownian diffusion in dynamically disordered thermal environments. J. Phys. Chem. B 121, 7204–7209 (2017).
    OpenUrl
  43. ↵
    1. S. Acharya,
    2. U. K. Nandi,
    3. S. Maitra Bhattacharyya
    , Fickian yet non-Gaussian behaviour: A dominant role of the intermittent dynamics. J. Chem. Phys. 146, 134504 (2017).
    OpenUrl
  44. ↵
    1. C. Beck,
    2. E. G. Cohen
    , Superstatistics. Physica A 322, 267–275 (2003).
    OpenUrlCrossRef
  45. ↵
    1. S. Hapca,
    2. J. W. Crawford,
    3. I. M. Young
    , Anomalous diffusion of heterogeneous populations characterized by normal diffusion at the individual level. J. R. Soc. Interface 6, 111–122 (2009).
    OpenUrlCrossRefPubMed
  46. ↵
    1. M. V. Chubynsky,
    2. G. W. Slater
    , Diffusing diffusivity: A model for anomalous, yet Brownian, diffusion. Phys. Rev. Lett. 113, 098302 (2014).
    OpenUrl
  47. ↵
    1. R. Jain,
    2. K. L. Sebastian
    , Diffusion in a crowded, rearranging environment. J. Phys. Chem. B 120, 3988–3992 (2016).
    OpenUrl
  48. ↵
    1. T. J. Lampo,
    2. S. Stylianidou,
    3. M. P. Backlund,
    4. P. A. Wiggins,
    5. A. J. Spakowitz
    , Cytoplasmic RNA-protein particles exhibit non-Gaussian subdiffusive behavior. Biophys. J. 112, 532–542 (2017).
    OpenUrl
  49. ↵
    1. A. V. Chechkin,
    2. F. Seno,
    3. R. Metzler,
    4. I. M. Sokolov
    , Brownian yet non-Gaussian diffusion: From superstatistics to subordination of diffusing diffusivities. Phys. Rev. X 7, 021002 (2017).
    OpenUrl
  50. ↵
    1. V. Sposini,
    2. A. V. Chechkin,
    3. F. Seno,
    4. G. Pagnini,
    5. R. Metzler
    , Random diffusivity from stochastic equations: Comparison of two models for Brownian yet non-Gaussian diffusion. New J. Phys. 20, 043044 (2018).
    OpenUrl
  51. ↵
    1. J. Ślęzak,
    2. R. Metzler,
    3. M. Magdziarz
    , Superstatistical generalised Langevin equation: Non-Gaussian viscoelastic anomalous diffusion. New J. Phys. 20, 023026 (2018).
    OpenUrl
  52. ↵
    1. A. Rahman,
    2. K. Singwi,
    3. A. Sjölander
    , Theory of slow neutron scattering by liquids. I. Phys. Rev. 126, 986 (1962).
    OpenUrl
  53. ↵
    1. A. Rahman
    , Correlations in the motion of atoms in liquid argon. Phys. Rev. 136, A405 (1964).
    OpenUrlCrossRef
  54. ↵
    1. T. M. Nieuwenhuizen,
    2. M. Ernst
    , Excess noise in a hopping model for a resistor with quenched disorder. J. Stat. Phys. 41, 773–801 (1985).
    OpenUrl
  55. ↵
    1. J. McLennan
    , Burnett coefficients and correlation functions. Phys. Rev. A 8, 1479 (1973).
    OpenUrl
  56. ↵
    1. I. De Schepper,
    2. H. Van Beyeren,
    3. M. Ernst
    , The nonexistence of the linear diffusion equation beyond Fick’s law. Physica 75, 1–36 (1974).
    OpenUrl
  57. ↵
    1. H. Zhang,
    2. J. F. Douglas
    , Glassy interfacial dynamics of Ni nanoparticles: Part I Colored noise, dynamic heterogeneity and collective atomic motion. Soft Matter 9, 1254–1265 (2013).
    OpenUrl
  58. ↵
    1. W.-S. Xu,
    2. J. F. Douglas,
    3. K. F. Freed
    , Influence of cohesive energy on relaxation in a model glass-forming polymer melt. Macromolecules 49, 8355–8370 (2016).
    OpenUrl
  59. ↵
    1. I. M. Sokolov,
    2. J. Klafter
    , From diffusion to anomalous diffusion: A century after Einstein’s Brownian motion. Chaos 15, 26103 (2005).
    OpenUrlPubMed
  60. ↵
    1. S. Chandrasekhar
    , Stochastic problems in physics and astronomy. Rev. Mod. Phys. 15, 1–89 (1943).
    OpenUrlCrossRef
  61. ↵
    1. J. Sung,
    2. R. J. Silbey
    , Counting statistics of single molecule reaction events and reaction dynamics of a single molecule. Chem. Phys. Lett. 415, 10–14 (2005).
    OpenUrl
  62. ↵
    1. H. Risken
    , The Fokker-Planck Equation (Springer, 1996), pp. 63–95.
  63. ↵
    1. R. Kubo
    , The fluctuation-dissipation theorem. Rep. Prog. Phys. 29, 255–284 (1966).
    OpenUrlCrossRef
  64. ↵
    1. J. L. Abascal,
    2. C. Vega
    , A general purpose model for the condensed phases of water: TIP4P/2005. J. Chem. Phys. 123, 234505 (2005).
    OpenUrlCrossRefPubMed
  65. ↵
    1. T. Odagaki
    , Non-ergodicity and non-Gaussianity in vitrification process. Prog. Theor. Phys. Suppl. 126, 9–12 (1997).
    OpenUrl
  66. ↵
    1. T. Odagaki,
    2. Y. Hiwatari
    , Stochastic model for the glass transition of simple classical liquids. Phys. Rev. A 41, 929–937 (1990).
    OpenUrlCrossRefPubMed
  67. ↵
    1. Y. He,
    2. S. Burov,
    3. R. Metzler,
    4. E. Barkai
    , Random time-scale invariant diffusion and transport coefficients. Phys. Rev. Lett. 101, 058101 (2008).
    OpenUrlCrossRefPubMed
  68. ↵
    1. P. Gallo,
    2. M. Rovere
    , Mode coupling and fragile to strong transition in supercooled TIP4P water. J. Chem. Phys. 137, 164503 (2012).
    OpenUrlCrossRefPubMed
  69. ↵
    1. T. Kawasaki,
    2. K. Kim
    , Identifying time scales for violation/preservation of Stokes-Einstein relation in supercooled water. Sci. Adv. 3, e1700399 (2017).
    OpenUrlFREE Full Text
  70. ↵
    1. S. Saito,
    2. B. Bagchi
    , Thermodynamic picture of vitrification of water through complex specific heat and entropy: A journey through “no man’s land”. J. Chem. Phys. 150, 054502 (2019).
    OpenUrl
  71. ↵
    1. C. Stanton,
    2. M. Nelkin
    , Random-walk model for equilibrium resistance fluctuations. J. Stat. Phys. 37, 1–16 (1984).
    OpenUrl
  72. ↵
    1. A. Y. Smirnov,
    2. A. A. Dubkov
    , Anomalous non-Gaussian diffusion in small disordered rings. Physica A 232, 145–161 (1996).
    OpenUrl
  73. ↵
    1. K. Krynicki,
    2. C. D. Green,
    3. D. W. Sawyer
    , Pressure and temperature dependence of self-diffusion in water. Faraday Discuss. Chem. Soc. 66, 199–208 (1978).
    OpenUrlCrossRef
  74. ↵
    1. G. E. Uhlenbeck,
    2. L. S. Ornstein
    , On the theory of the Brownian motion. Phys. Rev. 36, 823–841 (1930).
    OpenUrlCrossRef
  75. ↵
    1. W. Deng,
    2. E. Barkai
    , Ergodic properties of fractional Brownian-Langevin motion. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 79, 011112 (2009).
    OpenUrlCrossRefPubMed
  76. ↵
    1. T. Uneyama,
    2. T. Miyaguchi,
    3. T. Akimoto
    , Fluctuation analysis of time-averaged mean-square displacement for the Langevin equation with time-dependent and fluctuating diffusivity. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 92, 032140 (2015).
    OpenUrl
  77. ↵
    1. B. J. Berne,
    2. J. P. Boon,
    3. S. A. Rice
    , On the calculation of autocorrelation functions of dynamical variables. J. Chem. Phys. 45, 1086–1096 (1966).
    OpenUrlCrossRef
  78. ↵
    1. J. F. Douglas,
    2. J. B. Hubbard
    , Semiempirical theory of relaxation: Concentrated polymer solution dynamics. Macromolecules 24, 3163–3177 (1991).
    OpenUrlCrossRef
  79. ↵
    1. M. F. Shlesinger
    , Origins and applications of the Montroll-Weiss continuous time random walk. Eur. Phys. J. B 90, 93 (2017).
    OpenUrl
  80. ↵
    1. J. Sung,
    2. E. Barkai,
    3. R. J. Silbey,
    4. S. Lee
    , Fractional dynamics approach to diffusion-assisted reactions in disordered media. J. Chem. Phys. 116, 2338–2341 (2002).
    OpenUrl
  81. ↵
    1. J. Sung,
    2. R. J. Silbey
    , Exact dynamics of a continuous time random walker in the presence of a boundary: Beyond the intuitive boundary condition approach. Phys. Rev. Lett. 91, 160601 (2003).
    OpenUrlPubMed
  82. ↵
    1. K. Seki,
    2. M. Wojcik,
    3. M. Tachiya
    , Fractional reaction-diffusion equation. J. Chem. Phys. 119, 2165–2170 (2003).
    OpenUrl
  83. ↵
    1. B. Alder,
    2. T. Wainwright
    , Decay of the velocity autocorrelation function. Phys. Rev. A 1, 18 (1970).
    OpenUrlCrossRef
  84. ↵
    1. M. Ernst,
    2. E. Hauge,
    3. J. Van Leeuwen
    , Asymptotic time behavior of correlation functions. Phys. Rev. Lett. 25, 1254 (1970).
    OpenUrl
  85. ↵
    1. J. Dorfman,
    2. E. Cohen
    , Velocity correlation functions in two and three dimensions. Phys. Rev. Lett. 25, 1257 (1970).
    OpenUrl
  86. ↵
    1. B. Choi et al
    ., Nature of self-diffusion in two-dimensional fluids. New J. Phys. 19, 123038 (2017).
    OpenUrl
  87. ↵
    1. R. Zwanzig
    , Memory effects in irreversible thermodynamics. Phys. Rev. 124, 983–992 (1961).
    OpenUrlCrossRef
  88. ↵
    1. Y. R. Lim et al
    ., Quantitative understanding of probabilistic behavior of living cells operated by vibrant intracellular networks. Phys. Rev. X 5, 031014 (2015).
    OpenUrl
  89. ↵
    1. S. J. Park et al
    ., The Chemical Fluctuation Theorem governing gene expression. Nat. Commun. 9, 297 (2018).
    OpenUrl
  90. ↵
    1. H. Stehfest
    , Algorithm 368: Numerical inversion of Laplace transforms [D5]. Commun. ACM 13, 47–49 (1970).
    OpenUrlCrossRef
  91. ↵
    1. J. L. Abascal,
    2. C. Vega
    , Widom line and the liquid-liquid critical point for the TIP4P/2005 water model. J. Chem. Phys. 133, 234502 (2010).
    OpenUrlCrossRefPubMed
  92. ↵
    1. R. S. Singh,
    2. J. W. Biddle,
    3. P. G. Debenedetti,
    4. M. A. Anisimov
    , Two-state thermodynamics and the possibility of a liquid-liquid phase transition in supercooled TIP4P/2005 water. J. Chem. Phys. 144, 144504 (2016).
    OpenUrl
    1. N. Gov,
    2. A. G. Zilman,
    3. S. Safran
    , Hydrodynamics of confined membranes. Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 70, 011104 (2004).
    OpenUrlPubMed
PreviousNext
Back to top
Article Alerts
Email Article

Thank you for your interest in spreading the word on PNAS.

NOTE: We only request your email address so that the person you are recommending the page to knows that you wanted them to see it, and that it is not junk mail. We do not capture any email address.

Enter multiple addresses on separate lines or separate them with commas.
Transport dynamics of complex fluids
(Your Name) has sent you a message from PNAS
(Your Name) thought you would like to see the PNAS web site.
CAPTCHA
This question is for testing whether or not you are a human visitor and to prevent automated spam submissions.
Citation Tools
Transport dynamics of complex fluids
Sanggeun Song, Seong Jun Park, Minjung Kim, Jun Soo Kim, Bong June Sung, Sangyoub Lee, Ji-Hyun Kim, Jaeyoung Sung
Proceedings of the National Academy of Sciences Jun 2019, 116 (26) 12733-12742; DOI: 10.1073/pnas.1900239116

Citation Manager Formats

  • BibTeX
  • Bookends
  • EasyBib
  • EndNote (tagged)
  • EndNote 8 (xml)
  • Medlars
  • Mendeley
  • Papers
  • RefWorks Tagged
  • Ref Manager
  • RIS
  • Zotero
Request Permissions
Share
Transport dynamics of complex fluids
Sanggeun Song, Seong Jun Park, Minjung Kim, Jun Soo Kim, Bong June Sung, Sangyoub Lee, Ji-Hyun Kim, Jaeyoung Sung
Proceedings of the National Academy of Sciences Jun 2019, 116 (26) 12733-12742; DOI: 10.1073/pnas.1900239116
Digg logo Reddit logo Twitter logo Facebook logo Google logo Mendeley logo
  • Tweet Widget
  • Facebook Like
  • Mendeley logo Mendeley
Proceedings of the National Academy of Sciences: 116 (26)
Table of Contents

Submit

Sign up for Article Alerts

Article Classifications

  • Physical Sciences
  • Physics

Jump to section

  • Article
    • Abstract
    • 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
    • Discussion
    • Outlook
    • Methods
    • Acknowledgments
    • Footnotes
    • References
  • Figures & SI
  • Info & Metrics
  • PDF

You May Also be Interested in

Abstract depiction of a guitar and musical note
Science & Culture: At the nexus of music and medicine, some see disease treatments
Although the evidence is still limited, a growing body of research suggests music may have beneficial effects for diseases such as Parkinson’s.
Image credit: Shutterstock/agsandrew.
Scientist looking at an electronic tablet
Opinion: Standardizing gene product nomenclature—a call to action
Biomedical communities and journals need to standardize nomenclature of gene products to enhance accuracy in scientific and public communication.
Image credit: Shutterstock/greenbutterfly.
One red and one yellow modeled protein structures
Journal Club: Study reveals evolutionary origins of fold-switching protein
Shapeshifting designs could have wide-ranging pharmaceutical and biomedical applications in coming years.
Image credit: Acacia Dishman/Medical College of Wisconsin.
White and blue bird
Hazards of ozone pollution to birds
Amanda Rodewald, Ivan Rudik, and Catherine Kling talk about the hazards of ozone pollution to birds.
Listen
Past PodcastsSubscribe
Goats standing in a pin
Transplantation of sperm-producing stem cells
CRISPR-Cas9 gene editing can improve the effectiveness of spermatogonial stem cell transplantation in mice and livestock, a study finds.
Image credit: Jon M. Oatley.

Similar Articles

Site Logo
Powered by HighWire
  • Submit Manuscript
  • Twitter
  • Facebook
  • RSS Feeds
  • Email Alerts

Articles

  • Current Issue
  • Latest Articles
  • Archive

PNAS Portals

  • Anthropology
  • Chemistry
  • Classics
  • Front Matter
  • Physics
  • Sustainability Science
  • Teaching Resources

Information

  • Authors
  • Editorial Board
  • Reviewers
  • Librarians
  • Press
  • Site Map
  • PNAS Updates

Feedback    Privacy/Legal

Copyright © 2021 National Academy of Sciences. Online ISSN 1091-6490