Multiplicity of thermodynamic states of van der Waals gas in nanobubbles

Xu Tang,Hongguang Zhang,Zhenjiang Guo,Xianren Zhang*,Jing Li*,Dapeng Cao

State Key Laboratory of Organic-Inorganic Composites,Beijing University of Chemical Technology,Beijing 100029,China

Keywords:Nanobubbles Stability Thermodynamics Kinetics

ABSTRACT The gas-containing nanobubbles have attracted extensive attention due to their remarkable properties and extensive application potential.However,a number of fundamental aspects of nanobubbles,including thermodynamic states for the confined gas,remain still unclear.Here we theoretically demonstrate that the van der Waals (vdW) gases confined in nanobubbles exhibit a unique thermodynamic state of remarkably deviating from the bulk gas phase,and the state transition behavior due to the sizedependent Laplace pressure.In general,the vdW gas inside nanobubbles present multiple stable or transient states,where 0–2 states are for supercritical gas and 0–4 for subcritical gas.Our further analysis based on Rayleigh–Plesset equation and free energy determination indicates that the gas states in nanobubbles exhibits different levels of stability,from which the coexistence of multiple bubble states and microphase equilibrium between droplets and bubbles are predicted.This work provides insight to understand the thermodynamic states appeared for gas in nanobubbles.

1.Introduction

Nanobubbles (NBs) are gas-containing bubbles that have a size at the nanoscale[1,2].Depending on appearing on solid substrates or dispersing in bulk solution,they are called surface NBs or bulk NBs.The existence of surface NBs was first theoretically suggested in 1994[3],and then was confirmed by experimental observations shortly [4,5].Owing to their tiny size,NBs have large specific surface area and high internal pressure.Interestingly,both surface and bulk NBs demonstrate several unusual characteristics,such as unexpected stability with a lifetime of hours or days [6–9],long storage time,and strong adsorption capacity.These peculiar features of NBs lead to a wide range of promising applications,including medical imaging[10,11],detection in analytical chemistry[12],drug delivery [13,14],flotation [15,16],water-treatment techniques [17–20]and promoting growth of plants and animals [21].

Although the rapid development of various applications of surface NBs and bulk NBs,there are no in-depth studies on the thermodynamic states and microphase behaviors of the gaseous components confined in NBs.Owing to the finite size effect,gas packet encapsulated in NBs would experience tremendous Laplace pressure that is sensitive to bubble size.As shown below,the sizedependent Laplace pressure causes the occurrence of distinct thermodynamic states and even distinct microphases for gas molecules in NBs.

In this work,we used the classical van der Waals(vdW)gas,for which the bulk phase behavior is well known,to characterize the gas states and microphase behaviors in nanobubbles.This simple model gives us the opportunity to understand how typical features of gas states and microphases depend on NB size.The stability of resulting states was then analyzed with Helmholtz free energy and the Rayleigh–Plesset equation.Since various types of gas molecules can be capsulated in nanobubbles,we considered both nitrogen,a typical supercritical gas at room temperature,and C3F8,a typical subcritical gas as imaging contrast agent with lower acoustic impedance than blood and with improved ultrasound signaling[22–25].

2.Mathematical Model

We used a simple mathematical model to describe states of van der Waals (vdW) molecules encapsulated in nanobubbles immersed in liquid solvent.Here several assumptions are made to describe the thermodynamics and dynamics of gas-containing nanobubbles.(i) The bubbles have a spherical shape with only gas molecules included.(ii) The continuum assumption is applied for gas distribution in bulk NBs.For surface NBs with a much smaller height,Seddon et al.[26]argued that the gas in the nanobubbles is of Knudsen type.For bulk nanobubbles we studied here,however,the situation is different:for a typical size of 100 nm,a small Knudsen number,Kn=λ/R ≈0.038,is predicted,with λ the molecular mean free path and R the typical size of nanobubbles.Here Kn ?1 indicates the deviation from Knudsen gas behaviors.(iii)Laplace equation is applicable for nanobubbles because of its mechanical equilibrium nature.It is known that mechanical equilibrium can be reached much more quickly than diffusive equilibrium for gas crossing NB interfaces.iv)We ignored the interaction between neighboring nanobubbles.

For a nanobubble containing N gas molecules that is in mechanical equilibrium with the surrounding [see Fig.1(a)],the pressure inside the bubble Pgcan be determined with the Young-Laplace equation

Where P0is the external pressure from the surrounding,σ is the interfacial tension and R is bubble radius.

For the system in which the number of gas molecules inside the bubble N and the temperature T are kept constant,the pressure inside the bubble can be expressed by the van der Waals equation[27]:

where kBis the Boltzmann constant,V=4πR3/3 is bubble volume,NAis Avogadro constant.a=and b=kBTc/8Pcare the constants related to the critical temperature and critical pressure of the gas component (Tc,Pc).Substituting Eq.(2) into Eq.(1),we obtain the following relationship,

After determining the states of vdW gas with Eq.(3),we then analyzed their stability by introducing small perturbation to the equilibrium bubble size,i.e.,R →R+ΔR.The evolution dynamics for the deformed nanobubbles to reach equilibrium is described by the Rayleigh–Plesset (RP) equation [28,29],

where ρ is the liquid density and μ is the liquid viscosity.In all our calculations,we chose T=300 K,σ=0.072 N·m-1and μ=1.01×10-3Pa·s.The critical points for C3F8and nitrogen are,respectively (345.05 K,2.68 MPa) and (126.1 K,3.4 MPa).

Fig.1.Schematic illustration of nanobubble model and the variables used.(a) The system containing a nanobubble.(b,c)show respectively the reference state(b)and state studied (c) to calculate the free energy difference between them.

3.Results and Discussion

3.1.Appearance of multiple states for supercritical vdW gas in nanobubbles

Here we considered gas packet of nitrogen (in vdW model)encapsulated in NBs.At T=300 K,a temperature well above the critical temperature of nitrogen,vdW gas should behave as supercritical one.However,gas inside nanobubbles shows different state behaviors when compared to its bulk counterparts.

By solving Eq.(3),we can express the bubble radius R as a function of the external pressure P0at a given number of nitrogen molecules N.In general,the resulting solution can have 0–2 physical roots,depending on P0and N,as typically shown in Fig.2(a).At a positive external pressure,there is only a single solution for the bubble radius.With lowering the pressure to a negative value,a two-root region appears which is totally different from supercritical vdW gas in the bulk phase.Note that negative pressure is a common condition for various applications of gas nanobubbles,e.g.,negative pressure wound therapy and ultrasound imaging.When the external pressure further decreases,a threshold is reached,below which quick expansion of the bubble occurs,indicating the disappearance of stable nanobubbles.

3.2.Stability of nanobubbles from Rayleigh–Plesset equation

Then we analyzed the stability of those nanobubbles containing a given number of gas molecules by using RP equation.Note that at a specific number of gas molecules,the diffusive equilibrium for gas molecules crossing bubble interfaces may not achieve.The exchange of gas molecules across the interface may lead to the inequality of chemical potential,causing the NB to become unstable.However,as mechanical equilibrium can be reached much more quickly than diffusive equilibrium for gas crossing NB interfaces,we discussed here the NB stability within the time scale for gas reaching diffusive equilibrium.

To model the stability of a nanobubble under thermal perturbation,we first produced a deformed nanobubble which slightly departs from the equilibrium state from Eq.(3),namely Req(P0,N)→Req(P0,N)+ΔR.Then,RP equation was employed to describe the dynamic evolution of how the deformed bubble reaches its equilibrium state.When there exist multiple equilibrium states for a nanobubble at given(P0,N),the direction of evolution dynamics from RP equation (departure from or approaching one particular equilibrium state) can be used to distinguish whether the equilibrium state is stable or not.With Pgdetermined by Eq.(1),Fig.2(b) shows the typical evolution dynamics for the deformed bubbles denoted in Fig.2(a).

Fig.2.Typical states for supercritical gas in a nanobubble(a)and their stability analysis by RP equation(b).(a)Relationship between bubble radius and external pressure for the nanobubble containing a given number of nitrogen molecules of 106.The evolution direction of the deformed states denoted by ①–⑥,which is determined by RP equation,is also given in this figure with arrows.(b)Dynamic evolution process of the initially perturbed states ①–⑥,which are marked in Fig.2(a),is determined from the RP equation.The six initial states have respectively different radii R1 =70 nm,R2 =200 nm and R3 =400 nm,as well as different external pressures P0 =P1 (0.1 MPa),P0 =P2 (-0.5 MPa) and P0 =P3 (-1 MPa).

In general,Fig.2(a)shows the presence of three regimes for the states of supercritical gas confined inside nanobubbles,according to their stability of NBs:(i)compressed regime,in which deformed bubbles will grow into the state of stable nanobubbles;(ii)collapsing regime,in which deformed bubbles will collapse into the state of stable nanobubbles;(iii) expanding regime,in which deformed bubbles will expand spontaneously until macro-phase transition occurs.

3.3.Stability of nanobubbles from pressure equilibrium analysis and free energy analysis

Above,RP equation gives us a clue to the stability of various nanobubble states.Here we further analyze the stability of gascontaining nanobubbles by an approach based on pressure decomposition [31].For a bubble in mechanical equilibrium,Pl=Pg[Eq.(1)]should be satisfied,in which Pgcauses the bubble to expand,and Pl=P0+2σ/R makes the bubble shrink.For several states marked in Fig.2(a),the corresponding pressures Pgand Plat different external pressures P0=P1,P2,and P3are given in Fig.3(a),as a function of bubble size.

Fig.3(a) indicates that at a positive external pressure,e.g.,P0=0.1 MPa,there exists only one intersection for Pgand Pl,and the intersection corresponds to a gas state in stable equilibrium,as explained below.Suppose that the bubble radius is initially deformed to be larger than the equilibrium one,the contraction pressure (Pl=P0+2σ/R) would be larger than the expansion pressure (Pg),causing the bubble to shrink until both pressures reach equilibrium again.The same analysis can also be applied to the situation where the radius is smaller than the equilibrium value.Therefore,there exists only one equilibrium state for a given nanobubble at positive external pressure,and the state is in fact mechanically stable.

However,when the external pressure is reduced to negative value but still larger than the threshold pressuree.g.P0-=-0.5 MPa as shown in Fig.3(a),Pgand Plhave two intersections.In this case,the second state of the nanobubble appears,with a larger radius.Following the same analysis as done above for a positive pressure,we can infer that the smaller bubble withis mechanically stable.But for the larger nanobubble with R=the initial increase of nanobubble radius from the equilibrium value would cause a more rapid decrease of the contraction pressure (Pl=-P0+2σ/R) than the expansion pressure (Pg),leading to unbounded bubble growth.Therefore,the larger equilibrium bubble would easily depart from its equilibrium size under thermal perturbation,moving towards the smaller stable bubble or growing infinitely.

Fig.3.Stability of nanobubbles based on pressure decomposition and free energy analysis.(a)Graph of Pg (the left-hand side of Eq.(1))and Pl [namely P0 +2σ/R,the righthand side of Eq.(1)]as a function of nanobubble radius,under different values of external pressure P0 .(b)The free energy difference as a function bubble radius from Eq.(5)under different external pressures P0 .In both (a) and (b),the three different external pressure P0 are respectively set to P1 =0.1 MPa,P2 =-0.5 MPa and P3 =-1.0 MPa,corresponding to the pressures denoted in Fig.2(a).

When P0decreases further and becomes lower than the threshold pressure P0c,e.g.,P0=-1.0 MP in Fig.3(a),Plhas no intersection with Pg.In this case,Pgis always greater than Pl=P0+2σ/R since Laplace pressure decreases more significantly than Pgas bubble radius increases,causing unbounded growth of the bubble.In general,the stability analysis of nanobubbles bases on pressure equilibrium gives the same conclusions as those from the evolution dynamics for deformed nanobubbles.

Here we set a bubble with a size of 71 nm at the pressure of 0.1 MPa as the reference state.Note that the choice of the reference state does not affect the following stability analysis.

The resulting dependence of free energy change ΔF on the bubble size R is illustrated in Fig.3(b).The figure shows that at a positive external pressure,e.g.P0=P1=0.1 MPa,the free energy change first decreases to a minimum as R increases and then increases monotonically.The nanobubble featured with the minimum of free energy exactly corresponds to the equilibrium nanobubble obtained from Eq.(3),demonstrating clearly that for P0＞0 the equilibrium bubble states we observed in Fig.2(a) are stable.The free energy analysis also confirms that there exists only a single stable bubble under positive external pressure.

At mediated negative pressure ΔF would initially decrease to a minimum when R increases to a thresholdand then ΔF begins to increase until reaching a maximum that correspond to an unstable bubble with the larger size of[Fig.3(b)].This observation confirms that the equilibrium bubble with a smaller size ofis mechanically stable while that with the larger sizeis unstable.

When the external pressure becomes low enoughthe corresponding free energy difference decreases continuously as R increases.As a consequence,the bubble would expand infinitely.In general,the evolution dynamics,the pressure equilibrium analysis and free energy analysis give the same conclusions on the stability of various nanobubble states.The agreement confirms not only our conclusions on nanobubbles filled with supercritical gas,but also simplifies our discussion on nanobubbles with subcritical gas.

3.4.Effect of number of encapsulated nitrogen molecules on mechanical stability of nanobubbles

We also investigated the influence of the number of supercritical vdW molecules inside a nanobubble,N,on its mechanical stability.Again Eq.(3) was used to determine the relation between R and N for nanobubbles under various external pressures.

The extensive calculations are summarized in Fig.4.The figure clearly indicates that no matter what N is,there is only one stable bubble at a given positive external pressure (region I in Fig.4).Under negative external pressure (e.g.,P0=-0.1 MPa),however,there may exist two bubble states that are in mechanical equilibrium with the surrounding.The N-R curve in region II represents the stable branch of nanobubble states,while that in region III represents the unstable branch (Fig.4).With increasing N,the radius of the two states would gradually approach each other and merge at the threshold of NC,above which the nanobubble cannot stably exist.Fig.4 also demonstrates that as the external pressure decreases,the region surrounded by the both branches of the N-R curve would decrease gradually.

Therefore,we can divide the whole (R,N) plane into three regions.If the external pressure is positive,the nanobubble would equilibrate to the single stable state in region I.Region II represents a collection of states in which the stable nanobubble states can be found at a negative external pressure.Region III,on the other hand,shows the regime where unstable nanobubble states can be found.

3.5.Subcritical gas in nanobubbles:coexistence of multiple states

Above we discussed the state of supercritical gas(N2at normal temperature) confined in nanobubbles.We can expect that confined gas at subcritical condition would behave differently.In this work we used C3F8as an example to investigate states and state transitions for subcritical vdW gas encapsulated in nanobubbles.We chose C3F8also because it is widely used as a material for ultrasound contrast agent in medicine [22–25].

Fig.4.Nanobubble states in the plane (R,N).According to state properties,the whole plane is divided into three regions (denoted by different colors):Stable bubble states at positive external pressure are included in Region I (the bottomright corner).The stable branches of nanobubble states at negative external pressure are included in Region II,while those for unstable branches are included in Region III (the top-left corner).

Similarly,we combined the Young-Laplace equation and vdW equation to determine the state of confined C3F8as a function of external pressure P0and the number of confined C3F8molecules N[Eq.(3)].Fig.5 shows how nanobubble size changes with the external pressure.At small positive external pressure,P0-R curves have an undulating shape,and there exist 1–3 roots for the nanobubble radius at a given (P0,N) [see Fig.5(a)].The two spinodals are connected by the backward trajectory of unstable states of negative compressibility ((?P0/?R)N＞0),which are called internal states since these states would be subject to collapse if thermal fluctuations are present [33].The shape of isotherm,along with its similarity of the curve to van der Walls isotherm in bulk phase,leads us to conclude that the three roots correspond,respectively,to nanodroplet,unstable internal state,and nanobubble.The gas densities for these states also confirm the coexistence of both nanodroplet and nanobubbles at a given external pressure.

We also used R-P equation to confirm our conjecture on the mechanical stability of these states.The three states that are denoted in Fig.5(a) (at P0=1 MPa and N=109),which are intentionally chosen to depart from the equilibrium states,were used as the initial states for R-P equation calculations.The change of bubble radius with time is shown in Fig.5(c).The figure shows that at a given positive pressure,as expected,the deformed bubbles with a size between the minimal bubble (the nanodroplet) and the intermediately sized bubbles (the internal state) would shrink to the smaller equilibrium one(the nanodroplet),meaning that the nanodroplet state is stable.Instead,when the initial radius is between the size of intermediate bubble and that of the largest nanobubble,the bubble will evolve to the largest equilibrium nanobubble.In addition,when the bubble has an initial size larger than the largest equilibrium nanobubble,the bubble will again shrink to the equilibrium nanobubble,indicating that the bubble with the largest size is also stable.Therefore,the evolution of initial non-equilibrium bubbles from RP equation confirms our conjecture,i.e.,both the nanodroplet and nanobubble are stable,while the internal state with the intermediate bubble size is unstable mechanically.

It is more surprising that at negative external pressure,there may exist 0–4 root.For example,at -0.1 MPa and N=107,there are four roots [see Fig.5(b)],which represent,respectively,nanodroplet,internal state between the nanodroplet and nanobubble,nanobubble,and internal state for the macrobubble formation(gas-prompting cavitation).The first three roots are similar to these discussed above [see e.g.,Fig.5(a)]:they represent respectively nanodroplet,the internal state between nanodroplet and nanobubble,and nanobubble.The RP calculations also confirmed that they have the same stability as their counterparts under positive pressure:both nanodroplet and nanobubble are stable while the second root corresponds to an unstable state.For the fourth root,RP equation calculation shows that at this state the bubble is also unstable,corresponding to the internal state before gasprompting cavitation.With the external pressure continuously decreases,the number of bubble states would change to two,and eventually zero at which no bubble can exist [Fig.5(b)].

3.6.Subcritical gas in nanobubbles:microphase equilibrium between nanodroplets and nanobubbles

Fig.5.Typical states for subcritical gas in nanobubbles and their stability.(a,b) The relationship between external pressure and bubble radius for nanobubbles containing C3 F8 .(c) Dynamical evolution of initially deformed bubbles determined by RP equation.The initial states of the deformed bubbles correspond to the states ①–③that are marked in picture A,for which the radius are 500 nm,700 nm and 1000 nm respectively.In this figure,P0 =1 MPa.(d)The determination of microphase equilibrium between nanodroplets and nanobubbles according to Maxwell construction.The number of C3 F8 molecules was fixed to 109 in pictures(a),(c)and(d),while in picture(b)it was fixed to 107.

Hysteresis loop in Fig.5(a) indicates the occurrence of firstorder microphase transition between nanodroplet and nanobubble.The microphase equilibrium between nanodroplets and nanobubbles was determined by evaluating Gdroplet=Gbubbleorwith the integral of dG=VdP along the isotherm shown in Fig.5(a).This equality of Gibbs free energy(also chemical potential) for states is equivalent to the well-known equal-area Maxwell construction [see Fig.5(d)]:Requiring this integral to be zero is equivalent to the area bounded by the vertical line and the curve between nanodroplet and the unstable internal state equal to the area bounded by the vertical line and curve between the unstable state and the nanobubble.As an example,Fig.5(d)show that for the case of N=109,the chemical potential of the nanobubbles is equal to that for nanodroplets,μbubble=μdropletwhen P0=1.25 MPa.Therefore,for the given number of gas molecules encapsulated (for example N=109),nanodroplet and nanobubble are in equilibrium with each other as the external pressure reaches 1.25 MPa [the states denoted by b and e in Fig.5(d)].

As shown in Fig.5(d),the branch abc with a small bubble size corresponds to the density of condensed vdW fluid in the nanobubble,and the states on this branch are referred to as nanodroplets.The turnover point,c,corresponds to the point of spontaneous nanodroplet evaporation,at which the energy barrier separating the metastable nanodroplet and stable nanobubbles vanishes.The branch def with large size corresponds to nanobubble state.Similarly,the turnover point,d,corresponds to the true limits of stability of nanobubbles,and at this state the nanobubble must condense and jump onto the nanodroplet state spontaneously.The descending branch dc corresponds to internal states that are unstable.Therefore,at P0lower than the equilibrium external pressure (e.g.,P0=1.25 MPa for N=109),the nanobubble states are stable and the nanodroplet states are metastable.On the contrary,once P0exceeds the equilibrium pressure,nanodroplet states are stable and nanobubble states become metastable.

3.7.Coexistence of multiple states depends on the amount of gas confined and on the applied external pressure

Fig.6 gives the N-R relationship,which depends on the applied external pressure.Under the positive or zero external pressure,the number of bubble states changes from one to three as N increases,and then decreases to one [see Fig.6(a) and (b)].The range where three bubble states appear simultaneously at a given N corresponds to the coexistence of nanodroplets and nanobubbles.

Fig.6.The relationship between bubble radius and the number of C3 F8 molecules under different external pressures.The black solid lines indicate that the bubble states are stable,while the red dash lines indicate that the bubble states are unstable.

Under negative pressure,more states may occur [Fig.6(c)].There exist two bubble states for small N.As N increases continuously,the number of states that can coexist increases to four,and then decreases to two.When N increases to sufficiently large values,no equilibrium bubble state is found[Fig.6(c)].This is because for sufficiently large N,the bubble has a larger size in comparison to the critical nucleus for gas-containing cavitation transition,so that the bubble becomes unstable at a negative external pressure.

4.Conclusions

Gas-containing nanobubbles show a wide range of promising applications,while state behaviors of the confined gas molecules are little known.Owing to the finite size effect,gas packets encapsulated in nanobubbles would experience a tremendous Laplace pressure that depends on bubble size.In this work,the wellknown van der Waals (vdW) gas model was employed to describe the state of confined gas under the external pressure as well as the size-dependent Laplace pressure.We considered both nitrogen,a typical supercritical gas at room temperature,and C3F8,a typical subcritical gas that is extensively used as ultrasound contrast agent.As demonstrated with this simple model,the sizedependent pressure induces the appearance of different states and even additional phases for the confined vdW gas,in comparison with these presented in bulk gas phase.For supercritical vdW gas,0–2 states are found at a given external pressure,whereas for subcritical gas at most four equilibrium states are found.The stability of the obtained equilibrium states was then analyzed by using the Rayleigh–Plesset equation.Pressure decomposition approach and free energy analysis were also employed to confirm the conclusions on nanobubble stability.In general,the gas states in nanobubbles exhibits different levels of stability,and the coexistence of multiple bubble states and microphase equilibrium between droplets and bubbles are also predicted.

Our simple theoretical analysis indicates that the resulting states of vdW gas in nanobubbles have various levels of stability,due to the size-dependent Laplace pressure.However,we need to point out the stability discussed here never means that the bubbles are globally stable:similar to microbubbles for which the stability is of essential importance for their extensive applications,the diffusive equilibrium (as a result of diffusion of gas molecules crossing nanobubble interfaces) is not considered here.The exchange of gas molecules across bubble interface may lead to the inequality of chemical potential,causing the nanobubble to become eventually unstable.In other words,we discussed here the NB stability within the time scale for gas reaching diffusive equilibrium.The kind of stability is highly relevant to nanobubble applications.Normally the characteristic time for molecule exchange is much longer than that for mechanical equilibrium,and thus the states and state transitions discussed here can be readily achieved and thus can be experimentally tested.Furthermore,the unexpected nanobubble stability from experimental observations also demonstrates that the time scale for this kind of stability may be much longer than our prediction [6–9].We believe that our findings can serve as guideline for controlling nanobubble states by changing external pressure and bubble size.

Acknowledgements

This research was supported by the National Natural Science Foundation of China(21978007)and Fundamental Research Funds for the Central Universities (ZY1912).