Introduction

Envision a thin film of a molecular liquid exhibiting one or a combination of transitions: the liquid-liquid transition (LLT), where a semi-organized molecular structure is formed (liquid II); the liquid-solid transition (LST), in this case the liquid is transformed into a solid-like material and, lastly, the liquid-liquid-solid transition (LLST) defined as a combination of the two. When controlled, these transitions can be utilized in micro-electro-mechanical systems (MEMS), joint lubrication in biology and in micro-fluidics. Suspensions exhibit similar behaviors and the transitions play a role in paints, inks, cosmetics, pharmaceuticals and food1. There are also dense suspensions such as corn starch which exhibit jamming transitions and, in some cases, could allow you to run on their surface without sinking.

The question is: What happens during these transitions? Liquids brought to the transitional regime display collective (temporarily and spatially synchronized) motion of molecules and particles. Consequently, relaxation times are much longer from those in bulk2,3,4, viscosity increases by many orders of magnitude and, when sheared, confined liquids may experience smooth, stick-slip or chaotic responses5,6,7,8,9. The collective motion triggers dynamic heterogeneities with a supermolecular length ranging from a few up to ten molecular distances10,11,12,13. Heterogeneities of a similar kind are observed in various suspensions14 and, among them, in blood subjected to a transient flow15. We also know that strong stimuli drive viscoelastic liquids to the transitional state. An impact-activated solidification has been observed in dense suspensions16 and an aligned motion of molecules is detected in liquids subjected to shock17.

Molecular dynamics simulations provide excellent insight into the transitional phenomena18,19,20,21. The simulations are followed by thermodynamics-based models22,23 and minimalistic models24,25,26; the latter two are suitable for the replication of the observed behaviors. Also, the transitions are studied in the framework of non-Newtonian fluid dynamics27,28. As we noted earlier, the transitions emerge in over-constrained liquids where molecules lose their ability to move freely and, consequently, are forced to act in a collective manner.

In here, the transitional regime is the regime of our interest.

Our assertion is that a continuum-level approach can be useful in forecasting the transitions so long as the approach is brought close enough to the molecular scale. In our case, this is done by accounting for the relevant spatial and temporal fluctuations known as the dynamic heterogeneities29. Also, we assume that the substance in the liquid-like and solid-like states is different enough for the behaviors to be governed by independent constitutive equations. By comparing the two we derive criteria for the liquid-solid and liquid-liquid transitions in equilibrium and at steady state. The analysis prepares us for a more challenging task, namely the prediction of the LLSTs during transient flow processes. We illustrate the later in the example of blood subjected to an idealized cardiac cycle.

Results

Dynamic heterogeneity

Let's consider a viscoelastic liquid, where the monitored particle moves from its initial position {Xk} to a less than optimal position {xk}, Fig. 1. The “optimal” position refers to the thermodynamically most favorable (mean) positin {zk}. From this point of view, a collective motion of a few atoms in monatomic liquids (transit) can be considered a deviation from the expected trajectory30. In liquids, the current position of a molecule often diverges from the optimal position, but the trajectory must be physically admissible, i.e. it must be consistent with the constraints imposed by the conservation laws. In the framework of continuum, we expect to find the material particle (particle and its surroundings) in an acceptable position determined by the equations of motion , where the components of the current stress are σij, the particle acceleration is and, as usual, mass density is ρ. Any deviation from the thermodynamically optimal trajectory [ versus ] triggers perturbations in stress. Consider stress tractions plotted on a surface normal to the direction of flow {nk}. The tractions in the optimal and actual positions may not be the same and the difference is responsible for stress fluctuations

Stress in the reference (optimal) position is and in the actual position {xk} is σik. Using divergence theorem, the stress perturbations centered about become . The material length is understood as the dominant length and it captures the relevant spatial stretch of the stress gradient. Next, the stress gradient is replaced by the inertia term taken from the equations of motion, while volume V0 is reduced to a material point . As a result, the perturbations are simplified to . We assume that the tensor is symmetric and, then, we have . However, the symmetry restriction does not need to be enforced. The stress fluctuations are incorporated into the constitutive description of the liquid and the solid. In the transitional regime, the liquid is prone to shear and may experience changes in mass density. This behavior is described by the Maxwell-like viscoelastic model

where strain rate is . In this relation, elastic matrix is Cijkl and η is viscosity. Viscosity is determined by averages over a spectrum of relaxation times31. Usually, the viscous term in (2) is based on stress deviator alone. In this case, shear stress and pressure are viscous quantities32,33. Also, the elastic matrix includes contributions of bulk and shear moduli. The last term in (2) captures the contribution of the stress perturbations. The perturbations are added to the mean stress denoted as σij. In the constitutive relation, the parameter resembles the Reynolds number and, for this reason, we call it the material Reynolds number. We introduce the number for reasons discussed later. In here, kinematic viscosity is and υs is sound velocity. When the tensors and are not symmetric, we expect the non-symmetry would trigger perturbations in flow. The rate of mechanical work performed by the material is . The state function GL and the dissipation potential are and , respectively. We omit the contribution of heat flux. The flux in (2) may become a powerless quantity when the term is equal to zero. As suggested34, the powerless flux captures the contribution of hidden micro-scale dynamic events.

Figure 1
figure 1

Trajectory of a material point: starting from the position {Xk} the new optimal position would be in the point {zk}, but instead the particle travels to the actual position {xk}.

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The liquid is said to be converted to a viscoelastic solid-like material. In the simplest circumstance, the solid follows the Kelvin-Voigt behavior and includes the contribution of the micro-inertia described in (1), thus

The time span (t − t0) is taken to include the relevant history of the perturbations. This means that the material retains a short memory of the past history but this memory fades away beyond (t − t0)31,35. In here, mechanical work is . The state function is and the dissipation potential becomes . With the use of normality rules36 the dissipation potential captures viscous stress in (3). The contribution of the dynamic heterogeneity is linked to the relevant change in particle momentum. Under certain conditions the responses produced by the liquid (Eqn. 2) and the solid (Eqn. 3) become indistinguishable. This is what we call the liquid-liquid transitional state (liquid II).

LST- near-equilibrium scenario

We place the liquid (2) into a small container. Walls of the container restrict the motion of molecules and, in this manner, contribute to the increase of viscosity. In equilibrium, the fluctuations represent the primary response of the substance and, therefore, we omit the inertia terms in (2) and (3). In a one-dimensional setting, the nano-scale stress in the position {x} is calculated from the equation of motion , where σ and σz are stresses in the actual and optimal positions. At a larger (meso) scale, the stress becomes . Next, we separate the spatial and temporal terms in velocity and substitute the nano- and meso-stresses into the truncated constitutive equations (2) and (3). The nano-scale spatial perturbations in the liquid and the solid are Gaussian and, then, become harmonic at meso-scale , where is the magnitude of the perturbations. In the next step, we determine the temporal contribution υt(t). It turns out that the expression for υt(t) is scale independent (regardless whether it is the nano- or meso-scale), but υt(t) is different in the liquid and the solid

In here, the characteristic relaxation time is , while is sound velocity and, as before, is kinematic viscosity. The elastic constant is reduced to a single parameter C. A substance in the liquid and the solid state has very different properties and these properties become comparable only within the LST regime. The two expressions in (4) become identical when the material Reynolds number is equal to one. We conclude that the liquid-solid transition emerges when

The magnitude of kinematic viscosity increases as the size of the confinement becomes smaller7,8,37,38. As indicated, the transitional length in water, hexadecane, cyclohexane and other substances is in the range of six to ten molecular distances. When knowing sound velocity and viscosity (both measured at small scales), the predicted characteristic length is within the range observed in the experiments and predicted in molecular dynamics simulations.

LLT- steady flow

We have shown that spatial nano-fluctuations in equilibrium are Gaussian and become harmonic at meso-scale. It is suggested that steady shear makes the fluctuations non-Gaussian39 and, then, the fluctuations (1) trigger the liquid-liquid transition. We begin by enforcing conservation of mass and momentum , where the Cauchy stress is σ, velocity is and displacement is denoted as u. All the variables are expressed in terms of moving coordinate system , where the steady velocity is D. Consequently, we have , and strain is . Stress is derived directly from the conservation laws and is . In here, velocity is and ρ0 is the initial mass density.

As in the near equilibrium scenario, we predict the liquid-solid transition by comparing Eqns. 2 and 3; in here, both the relations include the contributions of the dynamic heterogeneities. The transition occurs when

Note that at the two material numbers are equal and, consequently, the two criteria (5) and (6) become identical. The transitional liquid (liquid II) should exhibit the properties described by (2) and (3). From the solution presented in Methods A, flow patterns exhibited by the liquid II are limited to

where u1 and u2 are constants. From (7) we see that the liquid II cannot be formed at . In all other situations, the liquid II must follow the script defined in (7). There are four scenarios:

  1. 1

    At , the substance is in its solid state (pastes, wet sands and other dense suspensions) exhibiting high resistance to flow (high viscosity). As a result, the material Reynolds number is small . The solid-liquid conversion is accomplished by applying a standing wave designed according to the protocol (7). The most known phenomenon of this kind is soil liquefaction40.

  2. 2

    One may envision another situation, where a viscoelastic liquid is subjected to the standing wave. In this manner, the liquid is forced to act as if it were a liquid II substance. Such experiments have been conducted14,41 and show the formation of microstructural patterns. We are interested in predicting the aggregation and disaggregation of red cells in blood not only under the steady state but also during transient flow42.

  3. 3

    Exponential flow of liquid II occurs at . Strongly driven liquids (under shock or impact) fit well the scenario. Often, it is assumed that shock wave has a sharp shock transition. In reality, the transition consists of several molecular layers of aligned molecules15 which (we predict) are organized within a thin membrane. The membrane travels through the material with shock velocity. Impact loading is also known to trigger the liquid-liquid conversions16.

  4. 4

    We predict that the liquid II behavior may emerge in liquids pushed from equilibrium and, then, allowed to relax according to the rules in (7). Supercooled liquids may fit the scenario, where a controllable decrease of temperature leads to an increase of viscosity, thus, affecting the substance's relaxation time21. We are not aware of any other experimental work done in this area.

LLST- short-lived transitions in blood during cardiac cycle

In simple terms, blood is a liquid tissue consisting of plasma and blood cells. On average 1 microliter of blood contains about red cells. Thus, in larger vessels (diameter 2 mm or larger), the number of cells is large enough for blood to become a homogeneous viscoelastic liquid. Blood viscosity and elasticity strongly depend on the actual blood composition, flow rate, shape and size of the blood vessels. A steady flow is fastest at the center of the vessel and slowest near the wall. This non-uniformity is linked to the buildup of wall shear stress. A pulsatile cycle produces considerably more complex flow patterns43,44. It is observed that vessel segments with low wall shear stress and oscillatory changes in flow direction appear to be at high risk for the development of various diseases and among them atherosclerosis and thrombosis. Atherosclerosis affects the inner lining of an artery and is characterized by plaque deposits that block the flow of blood. Thrombosis is the formation of clots capable of obstructing the flow of blood. Factors that affect blood viscosity45,46,47,48 are: hematocrit, red cell aggregation and deformation, plasma viscosity, concentration and size of low-density lipoproteins (LDL) and age.

Our objective is to forecast liquid-liquid-solid transitions in blood during cardiac cycle49. In our idealized scenario, the vessel is a rigid tube with inner radius . The tube is filled with blood and the system stays at rest. The derivations are based on Lagrange description, where strains are small, times are relatively short . The tube is large enough for the blood to remain in its liquid state . Next, the tube is rapidly moved from rest along its axis and kept in motion at constant velocity , Fig. 2. Blood slippages along the walls are not allowed. Whole blood in the state (L) is the viscoelastic liquid (2), where the temporal and spatial fluctuations are considered important. The blood in the transitional state (T) exhibits the properties of both the liquid (L) and the solid (S). The solid-like behavior is described in terms of the “local” Kelvin-Voigt model, where the stress fluctuations are omitted. The liquid (L), liquid II (T) and solid (S) are glued together, Fig. 2. The boundary of the liquid rL with respect to the boundary of the solid rS in the presence of the liquid II must be optimal in terms of the rate of work performed by the system , where the wall shear stress is and is shear stress. Expressions for the particle velocity and stress in each state are presented in Methods B. Blood viscosity and elasticity are determined for whole-blood (hematocrit 38%), where and . In each solution, the radius le of the vessel is equal to the significant stretch lc of the stress gradient. As stated earlier, the vessel is rapidly moved from rest and kept in motion at constant velocity . In terms of cardiac cycle this is the worst case scenario. The analysis is constructed for relatively large vessels, where the diameter is varying between three to six millimeters. These diameters correspond to the material Reynolds number in the range of two to four. Rapid departure from rest converts the liquid (L) into the liquid II (T) and the solid-like material (S), Fig. 2. In all the studied cases, clock is set to zero when all particles across the tube start sensing the motion, while the particle velocity at the center of the tube is still equal to zero. This setup properly replicates the velocity distribution in the tube at the moment of the blood flow reversal49. In the first example, at t = 0 the entire vessel contains blood in the transitional (T) and solid (S) states, Fig. 2. Gradually, blood is converted back to its liquid form (L) and at t ≈ τ0 the conversion is complete, where . There is a moment when the transitional liquid (T) disappears and a sharp liquid-solid interface emerges . At this point, the L-S interface migrates toward the tube walls. Stress tractions along the L-S interface are satisfied but velocity becomes discontinuous causing slippages. Such slippages have been observed in vessels near the vessel walls in a plasma layer of the thickness about 45.8 μm15,49. In larger vessels , the layers T and S are smaller and the conversion process is faster (Fig. 3). An indirect support for the LLSTs in blood at transient conditions is offered in ref. 15. The presence of the transitional and solid-like blood near the vessel walls is a concerning factor. We should note that an increase of blood viscosity and/or measurable decrease of elasticity may further aggravate the problem. Often, blood viscosity is considered the unifying indicator of cardiovascular diseases. It seems that the material Reynolds number would be a better predictor of cardiovascular disease risk.

Figure 2
figure 2

A rigid tube filled with blood is rapidly moved in axial direction (x) at constant velocity.

Initially, the velocity gradient triggers liquid-liquid-solid transitions but, later, blood is converted back to its liquid form. The liquid (L), transitional liquid (T) and solid (S) are marked in the tube and are plotted as a function of time. Velocity distributions are shown in 2-D and 3-D representations and are plotted for t = 0, 50, 100 and 150 ms.

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Figure 3
figure 3

In the upper right corner of the figure, we show the relaxation of the solid (S) and transitional (T) phases as a function of time for three vessels (D = 3.78, 4.54 and 6.04 nm).

The liquid II (T) and the solid (S) phases vanish faster as the size of the vessel increases. Also, the thickness of the T-S layer is plotted as a function of the vessel diameter. At the initial time of the flow reversal (t = 0), the T-S layer has a measurable thickness even in a very large vessel (D ≥ 20 mm). However, the size of the layer shrinks with the progression of time, as shown in instances (t = 10, 20, 30, 40 and 50 ms).

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Discussion

There are two aspects of the work worth noticing. First, our continuum level approach is adapted for a small scale analysis. We accomplish this by incorporating the spatial and temporal stress fluctuations into the material's description. Second, we view the substance either as a liquid-like or solid-like material. Thus, we diverge from the approaches where the liquid-like, solid-like and the transitional behaviors are constructed within a single mathematical framework. In this manner, we gain additional degree of freedom which we find necessary when describing the transitional processes. Consequently, we predict LLSTs during active flow processes (steady or transient); where in some circumstances the substance is hard driven. There are several mechanisms by which the transitions occur. Our criterion for the liquid-solid transformation works for liquids in small confinements. We predict microstructural reorganizations in liquids stimulated by standing waves, as we show that the standing waves are responsible for triggering liquefaction in dense suspensions. A synchronized motion of molecules (or particles in suspension) is forecasted in liquids subjected to an impact and/or shock loading. Lastly, we indicate that the liquid-liquid transitions should occur in liquids pushed away from equilibrium and, then, are allowed to relax according to a controllable scenario.

Methods

A. Transitional liquid (liquid II):

Suppose that the liquid-like and solid-like behaviors are indistinguishable. If so, we can substitute stress from (3) into the flow equation (2) and construct a general solution. First, we introduce an effective velocity

Next, we define an effective strain rate

The liquid II obeys the relations (2) and (3). We find that with the use of the new variables (A.1) and (A.2) the transitional liquid must follow the rule defined below

In here, is the initial velocity and is the initial strain rate. In addition, the material in the transitional state must adhere to the conservation laws. Under the condition of steady flow , the rules (A3) are

where stress is . Solution of the problem is presented in terms of displacements and stresses, namely

B. LLST in rigid vessel:

In a rigid vessel filled with blood, velocity and shear stress in the liquid (L) satisfy the equation of motion and are

There are two constants, namely and . Moreover, the velocity gradient in the center of the tube is always equal to zero , where .

The responses of the liquid II are constructed by solving the equation (A.3). Consequently, velocity and shear stress are

The transitional flow is determined in terms of three constants , and .

Lastly, the solution for the solid is

where and are constants. Boundary conditions for this problem are defined as follows:

We have seven constants and two time-dependent variables . The LLT and LLST boundaries are determined from the criterion of least action