Journal of Colloid and Interface Science 437 (2015) 259–269
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Journal of Colloid and Interface Science www.elsevier.com/locate/jcis
Modeling of microbubble dissolution in aqueous medium Sameer V. Dalvi ⇑, Jignesh R. Joshi Chemical Engineering, Indian Institute of Technology Gandhinagar, Chandkheda, Ahmedabad 382424, Gujarat, India
a r t i c l e
i n f o
Article history: Received 23 July 2014 Accepted 17 September 2014
Keywords: Microbubbles Dissolution Mass transfer Surface tension Elasticity
a b s t r a c t A mathematical model for microbubble dissolution in an aqueous medium containing dissolved gases is presented. None of the models available in the literature take into account the influence of shell elasticity (Es), variation in surface tension (r) at the gas–liquid interface and shell resistance (X) on the kinetics of microbubble dissolution. Moreover, values of these shell parameters are not known/available and hence arbitrary values for these variables have been assumed in many of the reports for estimation of dissolution kinetics. Therefore, in this work, a mathematical model is developed to describe microbubble dissolution which takes into account the effect of shell elasticity (Es), shell resistance (X), surface tension (r) and their variation, on the microbubble dissolution. The values of these shell parameters have then been estimated using the proposed model and the experimental data available in literature. The proposed model accurately predicts the experimental microbubble dissolution data using estimated values of shell parameters. Analysis of the results further show that the surface tension and shell resistances change drastically during the microbubble dissolution process and the variation in these parameters during the dissolution process is highly dependent on the shell elasticity which in turn affects the microbubble dissolution times. The methodology developed in this work can be used to estimate shell parameters for any microbubble formulation, to accurately predict in-vitro/in-vivo dissolution of microbubbles, and hence to design a microbubble system with desired characteristics and performance. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Microbubbles with size of up to 10 lm suspended in an aqueous medium are being used as contrast agents for ultrasonic imaging. These microbubbles are also emerging as carriers for targeted drug delivery. The gaseous core of these microbubbles is generally coated with a thin shell of materials such as proteins, lipids, polymers or surfactants [1]. The shell provides a barrier between gas and surrounding aqueous medium and thus adds resistance to the mass transfer of a gas from the microbubble to the surroundings. The microbubble dissolution dynamics and the time for microbubble dissolution (circulation persistence time) are very important for the viability of microbubbles for biomedical applications. There are several mathematical models available in the literature which try to capture the dynamics of the microbubbles dissolution mathematically. However, each model is based on certain assumptions which limit its application. A model suggested by Epstein and Plesset [2,3] does not take into account the shell resistance. It also assumes surface tension at the gas–liquid interface coated to be a constant throughout the microbubble ⇑ Corresponding author. Fax: +91 79 23972324. E-mail address: [email protected] (S.V. Dalvi). http://dx.doi.org/10.1016/j.jcis.2014.09.044 0021-9797/Ó 2014 Elsevier Inc. All rights reserved.
dissolution process. Subsequently, a model developed by Borden and Longo [4] assumes different values for shell resistance but also assumes shell to be inelastic material and hence the constant value for the surface tension. The models developed by Sarkar et al. [5] and Katiyar et al. [6] take into account shell elasticity but assume constant shell resistance. Further, the model developed by Azmin et al. [7] show that the shell resistance and surface tension changes during the dissolution process. However, it does not take into account the shell elasticity (Es) in the model. Thus, none of these models have accounted for variation in shell resistance, and variation in surface tension due to elastic shell material during the microbubble dissolution in a single model. Also, arbitrary values to many of the microbubble properties such as shell resistance, shell elasticity and the mass transfer resistance have been assumed in these reports. Therefore, a comprehensive model has been developed in this work which takes into account the effect of shell resistance (X), elasticity (Es), surface tension (r) and their variation on dissolution kinetics. The values of these unknown shell properties have been estimated by using the proposed model and the experimental dissolution data available in the literature [4,9]. These properties were further used to predict the microbubble dissolution behavior in an aqueous environment containing dissolved gases.
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Nomenclature Patm kb DAL DBL DCL DDL DEL CAl,1 CBl,1 CCl,1 CDl,1 CEl,1 ES H’ R0 f kG kL Xs,A
atmosphere pressure (Nm2) Boltzmann constant (J K1) coefficient of diffusivity of gas A in water (ms1) coefficient of diffusivity of gas B in water (ms1) coefficient of diffusivity of gas C in water (ms1) coefficient of diffusivity of gas D in water (ms1) coefficient of diffusivity of gas E in water (ms1) dissolved gas A concentrated far away from bubble in bulk aqueous medium (mol m3) dissolved gas B concentrated far away from bubble in bulk aqueous medium (mol m3) dissolved gas C concentrated far away from bubble in bulk aqueous medium (mol m3) dissolved gas D concentrated far away from bubble in bulk aqueous medium (mol m3) dissolved gas E concentrated far away from bubble in bulk aqueous medium (mol m3) elasticity of shell material (Nm1) Henry’s constant (unitless) initial radius of microbubble (m) level of saturation (unitless) local gas phase mass transfer coefficient (ms1) local liquid mass transfer coefficient (ms1) mass transfer resistance to gas A through microbubble shell (sm1)
The microbubbles are generally stored in an aqueous medium. This medium may contain a dissolved gas which is same as the gas used to make these microbubbles. In such a situation only a one way mass transfer of gas occurs from the core of a microbubble to the bulk of the aqueous medium. On the other hand, when a microbubble is injected in blood, a microbubble faces a multigas environment because of air dissolved in blood along with other gases. In such a case, two way mass transfer occurs where a gas from the microbubble core dissolves in blood and the gas dissolved in blood gets transferred into the microbubble. Therefore, first only a one way mass transfer across the gas–liquid interface was modeled. Subsequently, the model for one way mass transfer was extended to the dissolution of a microbubble in a multi-gas environment. The modified model accounts for the transfer of a gas from a microbubble core to the bulk of the aqueous medium as well as the flux of a gas from the bulk of the aqueous medium to the core of the microbubble.
mass transfer resistance to gas B through microbubble shell (sm1) mass transfer resistance to gas C through microbubble shell (sm1) mass transfer resistance to gas D through microbubble shell (sm1) mass transfer resistance to gas E through microbubble shell (sm1) microbubble surface tension at radius R (Nm1) molar concentration of gas A inside the microbubble molar concentration of gas B inside the microbubble molar concentration of gas C inside the microbubble molar concentration of gas D inside the microbubble molar concentration of gas E inside the microbubble molar flux across the interface (mol m2 s1) molar flux across the interface (mol m2 s1) molar flux across the interface (mol m2 s1) molar flux across the interface (mol m2 s1) molar flux across the interface (mol m2 s1) ostwald coefficient of gas A (unitless) ostwald coefficient of gas B (unitless) ostwald coefficient of gas C (unitless) ostwald coefficient of gas D (unitless) ostwald coefficient of gas E (unitless) overall mass transfer coefficient of liquid film (ms1)
XS,B XS,C XS,D XS,E
r(R) CAg CBg CCg CDg CEg NA NB NC ND NE LA LB LC LD LE KL
2.2. Model equations 2.2.1. Dissolution of a microbubble in the aqueous media containing a dissolved gas same as the one used to make the microbubble: One way mass transfer The schematic of a typical microbubble and concentration profiles inside and close vicinity of microbubble is shown in Fig. 1. It is assumed that the microbubble comprises of gas A and the surrounding medium also consists of gas A dissolved in it. The mole balance over the microbubble for a gas A yields 2rðRÞ 1 dnA 1 d Patm þ R 4 NA ¼ ¼ pR3 2 dt A dt 3 BT 4pR
!
where NA is molar flux across the interface, R is radius of microbubble, CAg is molar concentration of gas A inside the microbubble. Assuming gas inside the microbubble is to be ideal,
C Ag ¼
PAg BT
2. Model formulation 2.1. Model assumptions The model makes following simplifying assumptions: 1. There exist no convection in the microbubble storage medium or the aqueous medium and hence the mass transfer coefficient for liquid medium is calculated by assuming Sherwood (Sh) number to be 2. 2. Microbubble shell is completely hydrated and hence the gas solubility in the shell can be calculated using Henry’s law/Ostwald coefficient. 3. Gas in the microbubble core is an ideal gas. 4. Liquid phase controls the process of microbubble dissolution.
ð1Þ
Fig. 1. Schematic of a microbubble dissolving in an aqueous medium.
ð2Þ
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where PAg is the pressure inside the microbubble, B is universal gas constant (= 8.314 J/(mol K)), and T is temperature in K. PAg can be related to the microbubble surface tension using Laplace equation [5]
PAg ¼ P atm þ
2rðRÞ R
ð3Þ
where Patm is atmosphere pressure, r(R) is microbubble surface tension at radius R (N/m)
rðRÞ ¼ r0 þ Es ððR=R0 Þ2 1Þ
ð4Þ
where r0 is the surface tension at the gas–liquid interface (N/m), Es is the elasticity of shell material (N/m) and R0 is radius of microbubble at time t = 0. Substituting Eqs. (2)–(4) in (1), a following equation can be obtained
NA ¼
1 3BTRR20
dR 3Patm RR20 þ 4ðr0 Es ÞR20 þ 8Es R2 dt
ð5Þ
The molar flux of dissolving gas can also be expressed in terms of mass transfer coefficient as,
NA ¼ K L ðLA C Ag fC AL;1 Þ
ð6Þ
2 2 2 2 dR 3 LA fRR0 Patm þ 2ðr0 Es ÞR0 þ 2Es R g BTRR0 C AL;1 ¼ 1 dt þ DRAL 3Patm RR20 þ 4ðr0 Es ÞR20 þ 8Es R2 hA
ð13Þ
The shell permeability can be calculated as
hA ¼
1
ð14Þ
Xs;A
where Xs,A is the mass transfer resistance to gas A through a microbubble shell. Xs,A can be estimated using energy barrier model [8].
Xs;A ¼ Xn exp
pr2p ðr00 rÞ kb T
! ð15Þ
where Xn is the pre-exponential factor, kb is the Boltzmann constant, T is the absolute temperature and rp is the radius of the transferring species, r00 is the surface tension at the bare gas–liquid interface. Substituting Eq. (15) in (13), a following equation can be obtained
2 2 2 2 3 L fRR P þ 2ð r E ÞR þ 2E R g BTRR C A atm 0 s s AL;1 0 0 0 dR ¼ dt pr2p ðr00 rÞ þ DRAL 3Patm RR20 þ 4ðr0 Es ÞR20 þ 8Es R2 Xn exp k T b
where CAL,1 is a dissolved gas concentration far away from the bubble surface in bulk of the aqueous medium (mol/m3), KL is overall mass transfer coefficient based on liquid phase (m/s), LA is Ostwald coefficient of gas (dimensionless) and f is the saturation level of a gas A in the surrounding medium. The value of f is 1 if the surrounding medium is saturated with gas and it is zero for the completely unsaturated medium. Substituting Eqs. (2)–(4) in Eq. (6)
NA ¼
KL BTRR20
LA fRR20 Patm þ 2ðr0 Es ÞR20 þ 2Es R2 g BTRR20 C AL;1 ð7Þ
Comparing Eqs. (5) and (7)
ð8Þ The overall mass transfer coefficient based on the liquid phase can be estimated as
1 dnA 4pR2 dt
ð17Þ
where NA is mole flux of gas A mol/m2/s, R is radius of microbubble, nA is moles of gas A in the microbubble, Molar flux of A from the microbubble to the bulk of liquid medium can be expressed as
NA ¼ K L;A ðC Ag LA fC AL;1 Þ
ð18Þ
where CAg is molar concentration of gas A inside the microbubble.
ð9Þ C Ag ¼
where H’ is Henry’s constant, kG is local gas phase mass transfer coefficient, hA is shell permeability, kL is local liquid mass transfer coefficient. Assuming that the resistance in gas phase is negligible as compared to shell and liquid phase, Eq. (9) becomes,
1 1 1 ¼ þ K L hA kL
2.2.2. Dissolution of a microbubble in an aqueous medium containing dissolved gas other than the gas in microbubble core: Two way mass transfer Fig. 2 presents a schematic of a microbubble and the concentration profiles inside and close vicinity of microbubble dissolving in a multigas environment. It is assumed that the microbubble comprises of gas A and it is exposed to an aqueous medium which contains gas B dissolved in it. Taking balance for A over a microbubble
NA ¼
2 2 2 2 dR 3K L LA fRR0 Patm þ 2ðr0 Es ÞR0 þ 2Es R g BTRR0 C AL;1 ¼ dt 3Patm RR20 þ 4ðr0 Es ÞR20 þ 8Es R2
1 1 1 1 ¼ þ þ K L H0 kG hA kL
ð16Þ
3nA 4pR3
ð19Þ
nA is moles of gas A in Microbubble, R is radius of microbubble. CAL,1 is the concentration of a dissolved gas A far away from bubble in
ð10Þ
Also, assuming that no convections occur in the storage medium, the local liquid phase mass transfer coefficient (kL) can be estimated assuming Sh = 2 as follows:
kL ¼
DAL R
ð11Þ
Overall mass transfer coefficient then can be written as
KL ¼
1 hA
1 þ DRAL
Substituting Eq. (12) in Eq. (8)
ð12Þ Fig. 2. Schematic of a microbubble dissolving in a multigas environment.
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bulk of the aqueous medium (mol/m3), KL,A is overall mass transfer coefficient of gas A based on liquid phase (m/s), LA is Ostwald coefficient of gas and f is the saturation level of a dissolved gas in the surrounding medium. Substituting Eqs. (19) in (18).
NA ¼ K L;A
3nA LA 3
4p R
fC AL;1
ð20Þ
Comparing Eqs. (20) and (17)
dnA 3nA LA ¼ 4pR2 K L;A fC AL;1 dt 4pR3
ð21Þ
Overall mass transfer coefficient can be estimated as
K L;A ¼
1
The molar flux of B can be written as
NB ¼ K L;B ðfC BL;1 C Bg LB Þ
where CBg is molar concentration of gas A inside the microbubble. B C Bg ¼ 43n and nB is moles of gas B in microbubble, R is radius of pR3 microbubble, CBL,1 is a concentration of dissolved gas B in the bulk of aqueous medium (mol/m3), KL,B is overall mass transfer coefficient of gas B based on liquid phase (m/s), LB is Ostwald coefficient of gas (unitless) and f is the degree of saturation of the surrounding medium with respect to the dissolved gas. Comparing Eqs. (26) and (27),
dnB 3nB LB ¼ 4pR2 K L;B fC BL;1 dt 4pR3
K L;B ¼
where OS,A is mass transfer resistance to gas through microbubble shell. Substitute Eq. (22) in (21),
dnA 4pR2 3nA LA ¼ fC AL;1 dt Xs;A þ DRAL 4pR3
ð23Þ
1
ð29Þ
Xs;B þ DRBL
Substituting Eq. (29) in (28),
dnB 1 ¼ 4pR2 C Bg LB fC BL;1 dt Xs;B þ DR
pr2p ðr00 rÞ
ð30Þ
BL
Using energy theory for shell resistance
!
using energy barrier theory [8] for shell resistance,
Xs;A ¼ Xn;A exp
ð28Þ
Overall mass transfer coefficient can be written as
ð22Þ
Xs;A þ DRAL
ð27Þ
!
pr2p ðr00 rÞ
Xs;B ¼ Xn;B exp
ð31Þ
kb T
ð24Þ
kb T
Substituting Eq. (31) in (30), Substituting Eq. (24) in (23) 2
dnA 4pR 3nA LA pr2 ðr0 rÞ ¼ fC AL;1 3 p dt R 0 Xn;A exp þ DAL 4pR k T
ð26Þ
Eqs. (25) and (32) yield variation in number of moles of gas A and gas Bin the microbubble. To find out variation in microbubble radius with time overall balance can be written as
b
Similarly taking balances for gas B over a microbubble,
NB ¼
1 dnB 4pR2 dt 2
ð25Þ
where NB is mole flux of gas B mol/m /s, R is radius of microbubble, nB is moles of gas B in microbubble,
n;B
exp
1 pr2 ðr0 rÞ
3nB LB
dnB ¼ 4pR2 dt X
p
0
kb T
þ DRBL
4pR3
fC BL;1
ð32Þ
2r 4 3 Patm þ pR ¼ ðnA þ nB ÞBT 3 R
ð33Þ
Table 1 The estimated values of shell parameters and comparison of predicted values of shell resistances and the dissolution ties with the values reported in the literature. In the case of data presented below, the microbubble is dissolving in an aqueous medium containing dissolved gas same as the gas in the core of the microbubble (one way mass transfer). System
SDS coated air microbubble dissolving in air saturated water (f = 1) [9] SDS coated SF6 microbubble dissolving in an SF6 saturated water (f = 1) [9] Di12PC coated air microbubble dissolving in air saturated water (f = 0.833) [4] Di16PC coated air microbubble dissolving in air saturated water (f = 0.833) [4] Di18PC coated air microbubble dissolving in air saturated water (f = 0.833) [4] Di20PC coated air microbubble dissolving in air saturated water (f = 0.833) [4] Di22PC coated air microbubble dissolving in air saturated water (f = 0.833) [4] DMPC coated air microbubble dissolving in air saturated water (f = 0.833) [9] DPPC coated air microbubble dissolving in air saturated water (f = 0.833) [9] DSPC coated air microbubble dissolving in air saturated water (f = 0.833) [9] DAPC coated air microbubble dissolving in air saturated water (f = 0.833) [9] SDS-DPPC coated air microbubble dissolving in air saturated water (f = 1) [9] a
Es 105 (N/m) On 103 (s/m) r0 (N/m) Shell resistance (s/m)
tdiss (s)
Experimental Predicteda
Experimental Predicted % ARD in tdiss 50
1.673
1.01
0.0390
0
1018.27–1018.27
53.3
6.60
1.670
6.095
0.0390
0
26406.92–26426.52 386.4
499.2
29.19
120
0.4
0.0246
0
1290.68–1329.20
32.8
32.4
1.22
481
1.4
0.0249
5100 ± 2700
4484.02–5045.12
45.3
43.8
3.31
800
3.1
0.0247
8600 ± 5600
9978.11–12159.47
80.8
81.6
0.99
1000
6.2
0.0248
16,600 ± 5500 19906.96–31135.56 141
148
4.96
1810
6.8
0.0249
31,500 ± 7200 21779.55–27884.04 182
185
1.65
123
6.434
0.0246
13,070
20517.25–21144.04 103
94.8
7.96
410
8.0106
0.0249
14,664
25358.04–28033.62 129
120.1
6.90
900
15.02
0.0257
16,227
46625.15–58109.59 235
221.7
5.66
1030
17.01
0.0249
18,081
53846.18–69277.99 286
264.4
7.55
1.032
0.55
0.0283
0
1602.09–1602.50
79.7
3.24
77.2
A range in shell resistance (lowest and highest value) is given as the shell resistance increases during the microbubble dissolution.
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S.V. Dalvi, J.R. Joshi / Journal of Colloid and Interface Science 437 (2015) 259–269 Table 2 List of physical properties used in calculations. Physical property Coefficient of diffusivity of air in water Coefficient of diffusivity of SF6 in water Ostwald coefficient of air Ostwald coefficient of SF6 Temperature Collision diameter of air Collision diameter of SF6
3. Methodology of computation Value 9
2.05 10 m/s 1.2 109 m/s 1.71 102 5.4 103 295 K 3.58 1010 m 4.8 1010 m
3.1. Regression of shell properties from experimental data for microbubble dissolution A polynomial trend line was fitted to the experimental microbubble dissolution data (of Radius vs time) available in the literature [4,9]. Change in radius with time dR was then calculated dt by differentiating the polynomial equation with respect to time. To estimate the values of Es, On and r0, dR was calculated at t = 0, t = t dt and t = tdiss. This results into three nonlinear equations as shown below: At t = tdiss, R = 0, hence,
dR 3LA ¼ dt 2X exp pr2p Es n k T
ð35Þ
b
At t = 0, R = R0, hence,
3 2 2 3 dR 3 LA fR0 Patm þ 2ðr0 Es ÞR0 þ 2Es R0 g BTR0 C AL;1 ¼ dt Xn þ R0 3Patm R3 þ 4ðr0 Es ÞR2 þ 8Es R2 0
DAL
0
ð36Þ
0
At t = t, R = R, hence,
Fig. 3. Variation in dimensionless radius (R/R0) during dissolution of Di18PC coated air microbubble dissolving in water containing air dissolved at different saturation conditions (r0 = 0.0247 N/m, On = 3100 s/m, Es = 8 103 N/m).
3 LA fRR20 P atm þ 2ðr0 Es ÞR20 þ 2Es R2 g BTRR20 C AL;1 dR ¼ dt pr2p ðr00 rÞ R þ 3Patm RR20 þ 4ðr0 Es ÞR20 þ 8Es R2 Xn exp k T DAL b
ð37Þ The above three equations [Eqs. (35)–(37)] contain only three unknowns: Es, Xn and r0. The equations were solved simultaneously and the values of Es, Xn and r0 were estimated. 3.2. Estimating the dissolution kinetics for one way mass transfer i. Surface tension at the gas–liquid interface at any instant was calculated using Eq. (4) and pressure inside the microbubble was calculated using Eq. (3). ii. Shell resistance of gas was calculated by Eq. (15) and then overall mass transfer coefficient was estimated by Eq. (12). iii. Radius of the microbubble at every instant was then calculated using Eq. (16). 3.3. Estimating the dissolution kinetics for two way mass transfer i. Surface tension at the gas–liquid interface at any instant was calculated using Eq. (4) and pressure inside the microbubble was calculated using Eq. (3). ii. Shell resistance to gas diffusion was calculated using Eqs. (24) and (31) and then overall mass transfer coefficients were estimated using Eqs. (22) and (29). iii. The variation in moles of A at any instant was calculated as For gas A: 2
Fig. 4. Variation in (a) surface tension and (b) shell resistance with time at different saturation conditions during the dissolution of Di18PC coated air microbubble in air saturated water.
sþ1
nA
4pðRs Þ 3nsA LA ¼ nA Dt s fC AL;1 Rs XS;A þ DAL 4pðRs Þ3
!
s
ð38Þ
For gas B 2
The above equation after rearrangement becomes
4pP atm 3 8pr 2 R þ R ðnA þ nB Þ ¼ 0 3BT 3BT Eq. (34) can be solved to obtain values of R with time.
nBsþ1 ¼ nsB Dt ð34Þ
4pðRs Þ 3nsB LB fC BL;1 Rs XS;B þ DBL 4pðRs Þ3
! ð39Þ
iv. A microbubble radius at any instant was then estimated by solving the following equation.
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Table 3 Variation in dissolution time (tdiss), shell resistance and surface tension at the gas–liquid interface (coated with shell material) during dissolution of microbubbles filled with different gases and encapsulated in different shell materials in an aqueous medium containing varying degree of saturation of a dissolved gas. In the case of data presented below, the microbubble is dissolving in an aqueous medium containing dissolved gas same as the gas in the core of the microbubble (one way mass transfer). System
R0 (lm)
f
tdiss (s)
Range of XS,A (s/m) variation
Range of r (N/m) variation
SDS coated air microbubble dissolving in air saturated water
5
0 0.2 0.4 0.6 0.8 1
0.5 0.6 0.8 1.1 1.6 2.9
1018.27–1018.27 1018.27–1018.27 1018.27–1018.27 1018.27–1018.27 1018.27–1018.27 1018.27–1018.27
0.039–0.03898 0.039–0.03898 0.039–0.03898 0.039–0.03898 0.039–0.03898 0.039–0.03898
Di12PC coated air microbubble dissolving in air saturated water
5
0 0.2 0.4 0.6 0.8 1
0.7 0.8 0.1 1.2 1.6 2.4
1290.68–1329.20 1290.68–1328.80 1290.68–1329.40 1290.68–1329.03 1290.68–1329.16 1290.68–1329.20
0.0246–0.02341 0.0246–0.02342 0.0246–0.02340 0.0246–0.02341 0.0246–0.02341 0.0246–0.0234
Di16PC coated air microbubble dissolving in air saturated water
5
0 0.2 0.4 0.6 0.8 1
1.4 1.6 2 2.6 3.6 5.4
4484.02–5045.12 4484.02–5040.18 4484.02–5043.53 4484.02–5046.67 4484.02–5046.67 4484.02–5049.35
0.0249–0.02013 0.0249–0.02017 0.0249–0.02014 0.0249–0.02012 0.0249–0.0201 0.0249–0.02011
Di18PC coated air microbubble dissolving in air saturated water
5
0 0.2 0.4 0.6 0.8 1
3.4 4 4.75 5.9 7.85 11.8
9978.11–12159.47 9978.11–12158.34 9978.11–12159.03 9978.11–12159.48 9978.11–12159.02 9978.11–12159.31
0.0247–0.0167 0.0247–0.0167 0.0247–0.0167 0.0247–0.0167 0.0247–0.0167 0.0247–0.0167
Di20PC coated air microbubble dissolving in air saturated water
5
0 0.2 0.4 0.6 0.8 1
7.2 8.4 10 12.6 16.8 25.6
19906.96–31135.56 19906.96–31136.13 19906.96–31136.83 19906.96–31136.21 19906.96–31136.57 19906.96–31136.82
0.0248–0.0067 0.0248–0.0067 0.0248–0.0067 0.0248–0.0067 0.0248–0.0067 0.0248–0.0067
Di22PC coated air microbubble dissolving in air saturated water
5
0 0.2 0.4 0.6 0.8 1
7.9 9.3 11.3 14.3 19.6 31.3
21779.55–27884.04 21779.55–27884.01 21779.55–27881.04 21779.55–27884.32 21779.55–27883.83 21779.55–27885.55
0.0249–0.0149 0.0249–0.0149 0.0249–0.01491 0.0249–0.0149 0.0249–0.0149 0.0249–0.0149
SDS coated SF6 microbubble dissolving in an SF6 saturated water
5
0 0.2 0.4 0.6 0.8 1
23.4 27.4 33.2 42.4 59.2 105.2
26406.92–26426.52 26406.92–26426.52 26406.92–26426.52 26406.92–26426.52 26406.92–26426.52 26406.92–26426.52
0.039–0.03898 0.039–0.03898 0.039–0.03898 0.039–0.03898 0.039–0.03898 0.039–0.03898
DMPC PEG2000S coated air microbubble dissolving in air saturated water
5
0 0.2 0.4 0.6 0.8 1
5.9 7.0 8.7 11.7 17.8 42.0
20517.25–21144.04 20517.25–21143.91 20517.25–21143.79 20517.25–21144.05 20517.25–21143.98 20517.25–21143.97
0.0246–0.02337 0.0246–0.02337 0.0246–0.02337 0.0246–0.02337 0.0246–0.02337 0.0246–0.02337
DPPC PEG2000S coated air microbubble dissolving in air saturated water
5
0 0.2 0.4 0.6 0.8 1
7.71 9.20 11.44 15.26 23.35 55.86
25358.04–28033.62 25358.04–28033.62 25358.04–28033.62 25358.04–28033.62 25358.04–28033.62 25358.04–28033.62
0.0249–0.0208 0.0249–0.0208 0.0249–0.0208 0.0249–0.0208 0.0249–0.0208 0.0249–0.0208
DSPC PEG2000S coated air microbubble dissolving in air saturated water
5
0 0.2 0.4 0.6 0.8 1
15.10 18.11 22.67 30.27 46.32 108.24
46625.15–58109.59 46625.15–58109.59 46625.15–58109.59 46625.15–58109.59 46625.15–58109.59 46625.15–58109.59
0.0257–0.0167 0.0257–0.0167 0.0257–0.0167 0.0257–0.0167 0.0257–0.0167 0.0257–0.0167
DAPC PEG2000S coated air microbubble dissolving in air saturated water
5
0 0.2 0.4 0.6 0.8 1
18.04 21.60 27.02 36.29 56.18 136.57
53846.18–69277.99 53846.18–69277.99 53846.18–69277.99 53846.18–69277.99 53846.18–69277.99 53846.18–69277.99
0.0249–0.0146 0.0249–0.0146 0.0249–0.0146 0.0249–0.0146 0.0249–0.0146 0.0249–0.0146 (continued on next page)
265
S.V. Dalvi, J.R. Joshi / Journal of Colloid and Interface Science 437 (2015) 259–269 Table 3 (continued) System
R0 (lm)
f
tdiss (s)
Range of XS,A (s/m) variation
Range of r (N/m) variation
SDS-DPPC coated air microbubble dissolving in air saturated water
5
0 0.2 0.4 0.6 0.8 1
0.76 0.90 1.12 1.48 2.22 4.78
1602.09–1602.50 1602.09–1602.50 1602.09–1602.50 1602.09–1602.50 1602.09–1602.50 1602.09–1602.50
0.0283–0.02829 0.0283–0.02829 0.0283–0.02829 0.0283–0.02829 0.0283–0.02829 0.0283–0.02829
4pP atm ðsþ1Þ 3 8prðsÞ ðsþ1Þ 2 ðsþ1Þ ðsþ1Þ þ ðR Þ nA þ nB ¼0 R 3BT 3BT
ð40Þ
The time step (Dt) was varied as,
Dt ¼
R2 4Dmin
ð41Þ
where Dmin is the diffusivity of a gas with lowest diffusivity. 4. Results and discussions 4.1. Estimation of shell properties
Fig. 5. Variation in R/R0 for dissolution of SDS coated SF6 microbubble in water saturated with air at different saturation levels (r0 = 0.039 N/m, On = 1010 s/m, Es = 1.673 105 N/m).
7.00E-08
Molar Flux (kmol/m2 s)
The values of shell properties such as On, Es and r0 were estimated as discussed in Section 3.1. The experimental data on microbubble dissolution available in the literature [4,9] were used. The estimated values of On, Es and r0 are reported in Table 1. It can be observed from Table 1 that the elasticity of SDS as a shell material is much lower (at least by 2 orders of magnitude) than the elasticity of lipids (Di12PC to Di22PC). Therefore, SDS can be considered to be relatively inelastic as compared to lipids. Also, in case of lipids, the shell elasticity and shell resistance increases as the length of hydrophobic chain increases from Di12PC to Di22PC lipids. Table 1 presents a comparison of estimated shell resistances with the values available in the literature [4,9]. Comparison shows that the shell resistances reported in literature are lower than the values estimated in this work for most of the cases. The values reported in the literature were obtained without including shell elasticity or the variation in shell resistances with time which could be the reason for this disparity. The values of shell parameters thus estimated were further used for prediction of dissolution kinetics of microbubbles.
5.00E-08
NB 3.00E-08
1.00E-08
-1.00E-08
0.1
NA
1
10
100
1000
Time (seconds)
4.2. Kinetics of microbubble dissolution in an aqueous medium saturated/partially saturated with a gas used to make microbubble: One way mass transfer
Fig. 6. Variation in molar flux of SF6 (NA) and that of air (NB) with time during dissolution of SDS coated SF6 microbubble in water saturated with air (f = 1). Positive values of NB indicate influx of air into the microbubble whereas negative values of NA indicate efflux of SF6 from the microbubble.
Eq. (16) was solved numerically with initial condition of R = R0 at t = 0 to obtain variation in the radius of a microbubble dissolving in an aqueous medium. The values of On, Es and r0 presented in Table 1, were used along with the constants listed in Table 2 during the calculations. These calculations were performed in order to validate the proposed model and verify if the estimated values of shell properties can correctly predict the dynamics of microbubble dissolution. The predicted dissolution data match very well with the experimental dissolution data with R2 values ranging from 0.94 to 0.98 for all the systems studied in this work (See Figs. S1–S3). The values of dissolution times obtained also match closely with the experimental values of dissolution times for all the systems studied (Table 1). To explain the dynamics of the microbubble dissolution, a case of dissolution of Di18PC coated air microbubble in air saturated aqueous medium has been chosen. Fig. 3 presents predicted variation in R/R0 with time (solid lines) for Di18PC coated air bubble during its dissolution in an aqueous medium containing
air at different saturation levels. Experimental dissolution data for dissolution of a microbubble in an aqueous medium with air saturation level (f) of 0.833 (filled circles) are also presented along with the calculated dissolution profiles. It can be observed that the experimental data (for f = 0.833) matche very well with the predicted data (R2 = 0.98). Further, variation in surface tension (r) at the lipid coated air–water interface with time (Fig. 4a), and variation in lipid shell resistance (OSA) with time (Fig. 4b) has also been estimated. It was found that surface tension (r) decreases almost linearly with time (Fig. 4a) whereas the shell resistance increases with time (Fig. 4b). As microbubble dissolves, molecules of shell material get closer and get fully compressed which makes the microbubble shell increasingly rigid. This results in a decrease in surface tension with time. The compression of the molecules of shell material, on the other hand increases resistance for mass transfer of a gas due to a decrease in pore sizes in the compressed shell.
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Table 4 Variation in dissolution time (tdiss), shell resistance and surface tension at the gas–liquid interface (coated with shell material) during dissolution of microbubbles filled with SF6 and encapsulated in different shell materials in an aqueous medium containing varying degree of saturation of a dissolved air. In the case of data presented below, the microbubble dissolves in an aqueous medium containing a dissolved gas other than the core gas of the microbubble (two way mass transfer). Shell material
f
Variation in XS,A (s/m)
Variation in XS,B (s/m)
Variation in r (N/m)
SDS
0 0.2 0.4 0.6 0.8 1
6.73 7.27 8.02 9.19 11.46 21.18
6095–6099.48 6095–6099.48 6095–6099.48 6095–6099.48 6095–6099.48 6095–6099.48
1010–1010.41 1010–1010.41 1010–1010.41 1010–1010.41 1010–1010.41 1010–1010.41
0.039–0.03898 0.039–0.03898 0.039–0.03898 0.039–0.03898 0.039–0.03898 0.039–0.03898
Di12PC
0 0.2 0.4 0.6 0.8 1
2.69 3.03 3.52 4.34 6.17 16.86
971.50–1024.15 971.50–1024.15 971.50–1024.15 971.50–1024.15 971.50–1024.15 971.50–1024.15
400–411.92 400–411.92 400–411.92 400–411.92 400–411.92 400–411.92
0.0247–0.0235 0.0247–0.0235 0.0247–0.0235 0.0247–0.0235 0.0247–0.0235 0.0247–0.0235
Di16PC
0 0.2 0.4 0.6 0.8 1
5.18 5.66 6.34 7.43 9.72 21.69
3387.53–4185.40 3387.52–4185.41 3387.52–4185.42 3387.52–4185.432 3387.52–4185.45 3387.52–4185.55
1400–1574.794 1400–1574.796 1400–1574.798 1400–1574.8 1400–1574.804 1400–1574.825
0.0249–0.02009 0.0249–0.02009 0.0249–0.02009 0.0249–0.02009 0.0249–0.02009 0.0249–0.02009
Di18PC
0 0.2 0.4 0.6 0.8 1
10.03 10.74 11.70 13.17 16.02 30.32
7529.14–10704.18 7529.14–10704.18 7529.14–10704.18 7529.147–10704.18 7529.147–10704.18 7529.147–10704.18
3100–3770.192 3100–3770.192 3100–3770.192 3100–3770.192 3100–3770.193 3100–3770.193
0.0247–0.0167 0.0247–0.0167 0.0247–0.0167 0.0247–0.0167 0.0247–0.0167 0.0247–0.0167
Di20PC
0 0.2 0.4 0.6 0.8 1
21.08 22.31 23.95 26.32 30.56 50.61
16453.7–25542.76 16453.7–25542.77 16453.7–25542.79 16453.7–25542.8 16453.7–25542.82 16453.7–25542.91
6200–9653.897 6200–9653.903 6200–9653.911 6200–9653.919 6200–9653.931 6200–9653.987
0.0247–0.0066 0.0247–0.0066 0.0247–0.0066 0.0247–0.0066 0.0247–0.0066 0.0247–0.0066
Di22PC
0 0.2 0.4 0.6 0.8 1
25.21 26.41 27.93 30.02 33.52 52.17
15058.29–33380.47 15058.29–33380.51 15058.29–33380.55 15058.29–33380.61 15058.29–33380.68 15058.29–33381.03
6800–8684.771 6800–8684.773 6800–8684.775 6800–8684.778 6800–8684.782 6800–8684.798
0.0249–0.0149 0.0249–0.0149 0.0249–0.0149 0.0249–0.0149 0.0249–0.0149 0.0249–0.0149
DMPC PEG2000
0 0.2 0.4 0.6 0.8 1
15.88 17.12 18.85 21.56 27.03 58.34
16226.75–17137.29 16226.75–17137.32 16226.75–17137.36 16226.75–17137.43 16226.75–17137.52 16226.75–17137.99
6434–6632.40 6434–6632.40 6434–6632.41 6434–6632.43 6434–6632.45 6434–6632.55
0.0246–0.02337 0.0246–0.02337 0.0246–0.02337 0.0246–0.02337 0.0246–0.02337 0.0246–0.02337
DPPC PEG2000
0 0.2 0.4 0.6 0.8 1
21.37 22.83 24.82 27.84 33.58 62.38
20085.05–24431.41 20085.05–24431.44 20085.05–24431.49 20085.05–24431.54 20085.05–24431.61 20085.05–24431.94
8010.60–8932.85 8010.60–8932.85 8010.60–8932.86 8010.60–8932.87 8010.60–8932.89 8010.60–8932.95
0.0249–0.02049 0.0249–0.02049 0.0249–0.02049 0.0249–0.02049 0.0249–0.02049 0.0249–0.02049
DSPC PEG2000
0 0.2 0.4 0.6 0.8 1
43.02 45.52 48.78 53.41 61.42 96.40
37076.36–55279.11 37076.36–55280.11 37076.36–55281.44 37076.36–55283.27 37076.36–55286.13 37076.36–55297.27
15,020–18756.95 15,020–18757.14 15,020–18757.39 15,020–18757.74 15,020–18758.28 15,020–18760.38
0.0257–0.0167 0.0257–0.0167 0.0257–0.0167 0.0257–0.0167 0.0257–0.0167 0.0257–0.0167
DAPC PEG2000
0 0.2 0.4 0.6 0.8 1
52.56 55.36 58.97 64.04 72.59 110.44
42649.32–67397.25 42649.32–67397.32 42649.32–67397.39 42649.32–67397.48 42649.32–67397.59 42649.32–67398.00
17,010–21940.75 17,010–21940.76 17,010–21940.77 17,010–21940.79 17,010–21940.81 17,010–21940.88
0.0249–0.0146 0.0249–0.0146 0.0249–0.0146 0.0249–0.0146 0.0249–0.0146 0.0249–0.0146
tdiss (s)
Another interesting observation that could be made from Fig. 4a and b is that the rate of change of surface tension and shell resistance was found to be higher at the lower saturation levels as compared to the higher saturation levels of the aqueous medium. At lower saturation levels of the gas in the surrounding medium, the concentration gradient for the efflux of a gas from the core of the gas increases. This causes faster dissolution of a microbubble
leading to faster shrinkage/compression of shell material and hence higher rate of change of surface tension and the shell resistance. Table 3 presents variation in shell resistance and partial pressure of a gas in a microbubble during its dissolution for microbubbles made of different shell materials and different gases. It can be observed that for a material like SDS, the variation in shell
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Fig. 7. Variation in R/R0, and partial pressure of SF6 and that of air with time for different saturation conditions, for dissolution of Di18PC coated SF6 microbubble in water saturated with air (r0 = 0.0247 N/m, On = 3100 s/m, Es = 8 103 N/m).
resistance and surface tension with time is negligible (Table 3). However, the variation in surface tension and shell resistance to diffusion of gas is higher for lipids. This could be attributed to the higher elasticity of the lipid shells as compared to shell made of SDS.
4.3. Kinetics of microbubble dissolving in an aqueous medium containing dissolved gas other than the gas used to make the microbubble: Two way mass transfer The estimated values of shell parameters were further used for prediction of dissolution kinetics of SF6 filled microbubbles (R0 = 25 lm) coated with materials such as lipids (Di12PC to Di22C) and SDS in an aqueous medium containing air dissolved in it. In order to validate the proposed model for prediction of microbubble dissolution dynamics in a multigas environment, the predicted dissolution data for dissolution of SF6 filled SDS coated microbubble are compared with the dissolution data available in the literature [10]. Fig. 5 presents variation in non-dimensional radius (R/R0) with time for dissolution of SF6 filled SDS coated microbubble in water containing air dissolved at different saturation levels (f). The data predicted for dissolution of SF6 filled SDS coated microbubble in air saturated (f = 1) water were compared with the experimental dissolution data obtained from literature [10], as shown in Fig. 5. It can be observed from Fig. 5 that the model correctly captures dissolution kinetics of a microbubble. A microbubble growth phase is observed initially as a result of higher influx of air (NB) into the microbubble as compared to the efflux of SF6 (NA) out of the microbubble (Fig. 6). However, the mass transfer of air to the microbubble core (NB) increases the concentration of air in the microbubble core. This reduces the concentration gradient for the mass transfer of air from the surroundings to the microbubble core. Hence the flux of air (NB) decreases continuously with time during the microbubble dissolution (Fig. 6). A continuous decrease in influx of air (NB) along with the efflux of SF6 (NA) slows down the rate of increase in microbubble size during the growth phase. Eventually, the shrinking phase of a microbubble is triggered when the influx of air (NB) reduces to a very low value and even reverses the direction (shown in Fig. 6). The reversal in the direction of the flux of air (negative flux) indicates that the air now starts dissolving in the surrounding medium. This reversal happens as the reduction in microbubble size results in an increase in Laplace pressure inside the microbubble. Increase in Laplace pressure generates a concentration gradient for the transfer of air from
267
the microbubble core to the surroundings and hence air starts dissolving in the surrounding medium. A continuous decrease in the number of moles from the microbubble core due to the effluxes of SF6 as well as air, results in a decrease in the microbubble size to almost zero. It can also be observed from Fig. 5 that the microbubble dissolution behavior changes drastically with a decrease in saturation level (f) of the aqueous medium. At higher saturation levels such as f = 1, a microbubble dissolution exhibits a growth phase followed by a shrinking phase. However, as the value of f decreases (f < 1), the growth phase disappears and the microbubble only shrinks during the dissolution process. This is mainly because of a decreased amount of dissolved air in the aqueous medium with a decrease in f. A decrease in dissolved air in the aqueous medium reduces the influx of air to the microbubble and hence a gradual disappearance of the growth phase. In fact, a microbubble growth phase is not observed at lower values of f (such as f = 0). It can also be observed from Fig. 5 that the dissolution time (tdiss) of a microbubble increases as the saturation level (f) of the medium increases. An increase in the saturation level, increases the concentration of dissolved air in aqueous medium. Increase in the amount of dissolved air in the aqueous medium causes microbubble to grow, which otherwise would have only shrunk in the absence of any dissolved air in the surrounding medium (f = 0). The existence of additional growth phase along with shrinking phase increases the dissolution time of the microbubble in the surrounding medium. Table 4 presents predicted dissolution time, variation in shell resistances and surface tensions for dissolution of SF6 filled microbubbles made of various shell materials in an aqueous medium containing air at different saturation levels ( f). It can be observed that for SDS coated SF6 microbubble, surface tension and the shell resistances to the mass transfer of air and SF6 do not vary significantly during the process of microbubble dissolution. This could be attributed to a lower value of elasticity of SDS shell. However, in case of lipids, where elasticity values are higher than that for SDS, a significant variation in the shell resistance to gases and surface tension during microbubble dissolution has been observed (Table 4). Therefore, to illustrate the variation in shell resistances and surface tension during the microbubble dissolution, a case of dissolution of Di18PC coated SF6 microbubble in air saturated aqueous medium has been chosen. Fig. 7 presents the variation of dimensionless radius (R/R0) with time for different saturation conditions during dissolution of Di18PC coated SF6 microbubble in water containing dissolved air. As was observed for the dissolution of SDS coated SF6 microbubble, the time of dissolution was found to increase with increase in air saturation level of water for Di18PC microbubble as well. Fig. 8(a), (b) and (c) present variation in surface tensions and shell resistance with time during the process of microbubble dissolution. The surface tension (r) at the gas–liquid interface exhibits a maximum whereas the shell resistances to the mass transfer of SF6 (OSA) and air (OSB) exhibit minimum during the process of microbubble dissolution. During the growth phase, as microbubble grows, molecules of the shell material move away from each other and get decompressed. This reduces the rigidity of the shell resulting in a decrease in interaction among molecules of the shell and hence surface tension increases. During the shrinking phase however, as microbubble dissolves, the molecules of the shell material get closer and get fully compressed. The shell becomes increasingly rigid and hence surface tension decreases with time. On the shell resistances however, the decompression and compression of shell material during the growth and shrinking phase respectively has an exactly opposite effect. As the microbubble grows, the gap between the molecules of the shell material increases with time. This increases gas perme-
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Fig. 8. (a) Variation in surface tension with time, and (b and c) variation in shell resistance for SF6 and air with time at different saturation conditions for Di18PC coated air microbubble dissolving in air saturated water.
ability through shell material and hence shell resistance decreases. However, the molecules of shell material come closer as microbubble shrinks, resulting in a tighter packing and increased interaction among the shell molecules which reduces the defects in the shell structure and restricts the diffusion of gas molecules through the shell. This results in an increase in the resistance for the mass transfer of gas. 5. Conclusions A new model for the prediction of dissolution kinetics of a microbubble in an aqueous environment containing dissolved gases is presented. The process of microbubble dissolution and the dissolution time both are greatly affected by the shell properties such as surface tension (r) at the gas–liquid interface stabilized by the shell molecules, mass transfer resistance (X) offered by the shell material and the shell elasticity (Es). However, the reports available in the literature on the modeling of microbubble dissolution [2–7,10] present models which do not take into account the influence of above mentioned shell properties and their variation on microbubble dissolution process. Moreover, the values of the above mentioned shell parameters are not known/ available and hence arbitrary values for these variables have been assumed in many of the reports [2,3,5–7,10] for estimation of dissolution kinetics. Therefore, a new model has been presented in this work, which takes into account the effect of shell parameters and their variation on the microbubble dissolution. The values of these shell properties have also been estimated using the proposed model and the experimental dissolution data available in the literature [4,9]. The results show that the lipid shell is more elastic than
the shell made of SDS as elasticity of a SDS shell was found to be much lower (at least by 2 orders of magnitude) than the elasticity of lipids (Di12PC to Di22PC). Comparison of estimated shell resistances with the values reported in the literature [4,9] shows that the shell resistances reported in literature are lower than the values estimated in this work for most of the cases. The estimated values of shell parameters were further used for prediction of dissolution kinetics of microbubbles in an aqueous medium containing dissolved gases. The predicted dissolution data match very well with the experimental data with R2 values ranging from 0.94 to 0.98 for all the systems studied in this work. Further analysis of the results shows that the surface tension (r) decreases almost linearly with time and the shell resistance increases with time during the microbubble dissolution in case of a one way mass transfer of a gas (i.e. only from microbubble core to the surroundings). However, during the microbubble dissolution in a multigas environment where a two way mass transfer occurs (i.e. from microbubble core to the surroundings and from surrounding medium to the microbubble core), the surface tension at the gas–liquid interface exhibits a maximum whereas the shell resistances to the mass transfer of gas (OSA) and air (OSB) exhibit minimum. It was also found that the variation in surface tension and shell resistances during the dissolution process is highly dependent on the shell elasticity. The lower shell elasticity was found to result in a negligible variation in the shell resistance and surface tension (as observed in the case of SDS) whereas higher shell elasticity was found to result in a large variation in surface tension and shell resistance during the microbubble dissolution (as observed in case of lipids). This is in contrast to the reports available in the literature [2–7,10] where surface tension and shell resistance was assumed
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to be constant and no dependence of these shell properties on shell elasticity was taken into account. The model presented in this work, can further be extended to predict the circulation persistence time of microbubbles injected into a blood stream and to predict the shelf stability of any microbubbles formulation. Moreover, the methodology presented in this work can be used for the purpose of designing a microbubble system with desired characteristics and performance. Acknowledgments Authors gratefully acknowledge the financial support from the Department of Biotechnology of Ministry of Science and Technology, and Ministry of Human Resource Development of Government of India, and Indian Institute of Technology, Gandhinagar (IITGN).
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Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.jcis.2014.09.044. References [1] [2] [3] [4] [5] [6] [7]
S. Sirsi, M. Borden, Bubble Sci. Eng. Technol. 1 (1–2) (2009) 3–17. S.P. Epstein, J. Chem. Phys. 18 (11) (1950) 1505–1509. P.B. Duncan, D. Needham, Langmuir 20 (7) (2004) 2567–2578. M.A. Borden, M.L. Longo, Langmuir 18 (24) (2002) 9225–9233. K. Sarkar, A. Katiyar, P. Jain, Ultrasound Med. Biol. 35 (8) (2009) 1385–1396. A. Katiyar, K. Sarkar, P. Jain, J. Colloid Interface Sci. 336 (2) (2009) 519–525. M. Azmin, G. Mohamedi, M. Edirisinghe, E.P. Stride, Mater. Sci. Eng. C 32 (8) (2012) 2654–2658. [8] J.J. Kwan, M.A. Borden, Adv. Colloid Interface Sci. 183–184 (2012) 82–99. [9] M.M. Lozano, M.L. Longo, Langmuir 25 (6) (2009) 3705–3712. [10] J.J. Kwan, M.A. Borden, Langmuir 26 (9) (2010) 6542–6548.
Contents lists available at ScienceDirect
Journal of Colloid and Interface Science www.elsevier.com/locate/jcis
Modeling of microbubble dissolution in aqueous medium Sameer V. Dalvi ⇑, Jignesh R. Joshi Chemical Engineering, Indian Institute of Technology Gandhinagar, Chandkheda, Ahmedabad 382424, Gujarat, India
a r t i c l e
i n f o
Article history: Received 23 July 2014 Accepted 17 September 2014
Keywords: Microbubbles Dissolution Mass transfer Surface tension Elasticity
a b s t r a c t A mathematical model for microbubble dissolution in an aqueous medium containing dissolved gases is presented. None of the models available in the literature take into account the influence of shell elasticity (Es), variation in surface tension (r) at the gas–liquid interface and shell resistance (X) on the kinetics of microbubble dissolution. Moreover, values of these shell parameters are not known/available and hence arbitrary values for these variables have been assumed in many of the reports for estimation of dissolution kinetics. Therefore, in this work, a mathematical model is developed to describe microbubble dissolution which takes into account the effect of shell elasticity (Es), shell resistance (X), surface tension (r) and their variation, on the microbubble dissolution. The values of these shell parameters have then been estimated using the proposed model and the experimental data available in literature. The proposed model accurately predicts the experimental microbubble dissolution data using estimated values of shell parameters. Analysis of the results further show that the surface tension and shell resistances change drastically during the microbubble dissolution process and the variation in these parameters during the dissolution process is highly dependent on the shell elasticity which in turn affects the microbubble dissolution times. The methodology developed in this work can be used to estimate shell parameters for any microbubble formulation, to accurately predict in-vitro/in-vivo dissolution of microbubbles, and hence to design a microbubble system with desired characteristics and performance. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Microbubbles with size of up to 10 lm suspended in an aqueous medium are being used as contrast agents for ultrasonic imaging. These microbubbles are also emerging as carriers for targeted drug delivery. The gaseous core of these microbubbles is generally coated with a thin shell of materials such as proteins, lipids, polymers or surfactants [1]. The shell provides a barrier between gas and surrounding aqueous medium and thus adds resistance to the mass transfer of a gas from the microbubble to the surroundings. The microbubble dissolution dynamics and the time for microbubble dissolution (circulation persistence time) are very important for the viability of microbubbles for biomedical applications. There are several mathematical models available in the literature which try to capture the dynamics of the microbubbles dissolution mathematically. However, each model is based on certain assumptions which limit its application. A model suggested by Epstein and Plesset [2,3] does not take into account the shell resistance. It also assumes surface tension at the gas–liquid interface coated to be a constant throughout the microbubble ⇑ Corresponding author. Fax: +91 79 23972324. E-mail address: [email protected] (S.V. Dalvi). http://dx.doi.org/10.1016/j.jcis.2014.09.044 0021-9797/Ó 2014 Elsevier Inc. All rights reserved.
dissolution process. Subsequently, a model developed by Borden and Longo [4] assumes different values for shell resistance but also assumes shell to be inelastic material and hence the constant value for the surface tension. The models developed by Sarkar et al. [5] and Katiyar et al. [6] take into account shell elasticity but assume constant shell resistance. Further, the model developed by Azmin et al. [7] show that the shell resistance and surface tension changes during the dissolution process. However, it does not take into account the shell elasticity (Es) in the model. Thus, none of these models have accounted for variation in shell resistance, and variation in surface tension due to elastic shell material during the microbubble dissolution in a single model. Also, arbitrary values to many of the microbubble properties such as shell resistance, shell elasticity and the mass transfer resistance have been assumed in these reports. Therefore, a comprehensive model has been developed in this work which takes into account the effect of shell resistance (X), elasticity (Es), surface tension (r) and their variation on dissolution kinetics. The values of these unknown shell properties have been estimated by using the proposed model and the experimental dissolution data available in the literature [4,9]. These properties were further used to predict the microbubble dissolution behavior in an aqueous environment containing dissolved gases.
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Nomenclature Patm kb DAL DBL DCL DDL DEL CAl,1 CBl,1 CCl,1 CDl,1 CEl,1 ES H’ R0 f kG kL Xs,A
atmosphere pressure (Nm2) Boltzmann constant (J K1) coefficient of diffusivity of gas A in water (ms1) coefficient of diffusivity of gas B in water (ms1) coefficient of diffusivity of gas C in water (ms1) coefficient of diffusivity of gas D in water (ms1) coefficient of diffusivity of gas E in water (ms1) dissolved gas A concentrated far away from bubble in bulk aqueous medium (mol m3) dissolved gas B concentrated far away from bubble in bulk aqueous medium (mol m3) dissolved gas C concentrated far away from bubble in bulk aqueous medium (mol m3) dissolved gas D concentrated far away from bubble in bulk aqueous medium (mol m3) dissolved gas E concentrated far away from bubble in bulk aqueous medium (mol m3) elasticity of shell material (Nm1) Henry’s constant (unitless) initial radius of microbubble (m) level of saturation (unitless) local gas phase mass transfer coefficient (ms1) local liquid mass transfer coefficient (ms1) mass transfer resistance to gas A through microbubble shell (sm1)
The microbubbles are generally stored in an aqueous medium. This medium may contain a dissolved gas which is same as the gas used to make these microbubbles. In such a situation only a one way mass transfer of gas occurs from the core of a microbubble to the bulk of the aqueous medium. On the other hand, when a microbubble is injected in blood, a microbubble faces a multigas environment because of air dissolved in blood along with other gases. In such a case, two way mass transfer occurs where a gas from the microbubble core dissolves in blood and the gas dissolved in blood gets transferred into the microbubble. Therefore, first only a one way mass transfer across the gas–liquid interface was modeled. Subsequently, the model for one way mass transfer was extended to the dissolution of a microbubble in a multi-gas environment. The modified model accounts for the transfer of a gas from a microbubble core to the bulk of the aqueous medium as well as the flux of a gas from the bulk of the aqueous medium to the core of the microbubble.
mass transfer resistance to gas B through microbubble shell (sm1) mass transfer resistance to gas C through microbubble shell (sm1) mass transfer resistance to gas D through microbubble shell (sm1) mass transfer resistance to gas E through microbubble shell (sm1) microbubble surface tension at radius R (Nm1) molar concentration of gas A inside the microbubble molar concentration of gas B inside the microbubble molar concentration of gas C inside the microbubble molar concentration of gas D inside the microbubble molar concentration of gas E inside the microbubble molar flux across the interface (mol m2 s1) molar flux across the interface (mol m2 s1) molar flux across the interface (mol m2 s1) molar flux across the interface (mol m2 s1) molar flux across the interface (mol m2 s1) ostwald coefficient of gas A (unitless) ostwald coefficient of gas B (unitless) ostwald coefficient of gas C (unitless) ostwald coefficient of gas D (unitless) ostwald coefficient of gas E (unitless) overall mass transfer coefficient of liquid film (ms1)
XS,B XS,C XS,D XS,E
r(R) CAg CBg CCg CDg CEg NA NB NC ND NE LA LB LC LD LE KL
2.2. Model equations 2.2.1. Dissolution of a microbubble in the aqueous media containing a dissolved gas same as the one used to make the microbubble: One way mass transfer The schematic of a typical microbubble and concentration profiles inside and close vicinity of microbubble is shown in Fig. 1. It is assumed that the microbubble comprises of gas A and the surrounding medium also consists of gas A dissolved in it. The mole balance over the microbubble for a gas A yields 2rðRÞ 1 dnA 1 d Patm þ R 4 NA ¼ ¼ pR3 2 dt A dt 3 BT 4pR
!
where NA is molar flux across the interface, R is radius of microbubble, CAg is molar concentration of gas A inside the microbubble. Assuming gas inside the microbubble is to be ideal,
C Ag ¼
PAg BT
2. Model formulation 2.1. Model assumptions The model makes following simplifying assumptions: 1. There exist no convection in the microbubble storage medium or the aqueous medium and hence the mass transfer coefficient for liquid medium is calculated by assuming Sherwood (Sh) number to be 2. 2. Microbubble shell is completely hydrated and hence the gas solubility in the shell can be calculated using Henry’s law/Ostwald coefficient. 3. Gas in the microbubble core is an ideal gas. 4. Liquid phase controls the process of microbubble dissolution.
ð1Þ
Fig. 1. Schematic of a microbubble dissolving in an aqueous medium.
ð2Þ
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where PAg is the pressure inside the microbubble, B is universal gas constant (= 8.314 J/(mol K)), and T is temperature in K. PAg can be related to the microbubble surface tension using Laplace equation [5]
PAg ¼ P atm þ
2rðRÞ R
ð3Þ
where Patm is atmosphere pressure, r(R) is microbubble surface tension at radius R (N/m)
rðRÞ ¼ r0 þ Es ððR=R0 Þ2 1Þ
ð4Þ
where r0 is the surface tension at the gas–liquid interface (N/m), Es is the elasticity of shell material (N/m) and R0 is radius of microbubble at time t = 0. Substituting Eqs. (2)–(4) in (1), a following equation can be obtained
NA ¼
1 3BTRR20
dR 3Patm RR20 þ 4ðr0 Es ÞR20 þ 8Es R2 dt
ð5Þ
The molar flux of dissolving gas can also be expressed in terms of mass transfer coefficient as,
NA ¼ K L ðLA C Ag fC AL;1 Þ
ð6Þ
2 2 2 2 dR 3 LA fRR0 Patm þ 2ðr0 Es ÞR0 þ 2Es R g BTRR0 C AL;1 ¼ 1 dt þ DRAL 3Patm RR20 þ 4ðr0 Es ÞR20 þ 8Es R2 hA
ð13Þ
The shell permeability can be calculated as
hA ¼
1
ð14Þ
Xs;A
where Xs,A is the mass transfer resistance to gas A through a microbubble shell. Xs,A can be estimated using energy barrier model [8].
Xs;A ¼ Xn exp
pr2p ðr00 rÞ kb T
! ð15Þ
where Xn is the pre-exponential factor, kb is the Boltzmann constant, T is the absolute temperature and rp is the radius of the transferring species, r00 is the surface tension at the bare gas–liquid interface. Substituting Eq. (15) in (13), a following equation can be obtained
2 2 2 2 3 L fRR P þ 2ð r E ÞR þ 2E R g BTRR C A atm 0 s s AL;1 0 0 0 dR ¼ dt pr2p ðr00 rÞ þ DRAL 3Patm RR20 þ 4ðr0 Es ÞR20 þ 8Es R2 Xn exp k T b
where CAL,1 is a dissolved gas concentration far away from the bubble surface in bulk of the aqueous medium (mol/m3), KL is overall mass transfer coefficient based on liquid phase (m/s), LA is Ostwald coefficient of gas (dimensionless) and f is the saturation level of a gas A in the surrounding medium. The value of f is 1 if the surrounding medium is saturated with gas and it is zero for the completely unsaturated medium. Substituting Eqs. (2)–(4) in Eq. (6)
NA ¼
KL BTRR20
LA fRR20 Patm þ 2ðr0 Es ÞR20 þ 2Es R2 g BTRR20 C AL;1 ð7Þ
Comparing Eqs. (5) and (7)
ð8Þ The overall mass transfer coefficient based on the liquid phase can be estimated as
1 dnA 4pR2 dt
ð17Þ
where NA is mole flux of gas A mol/m2/s, R is radius of microbubble, nA is moles of gas A in the microbubble, Molar flux of A from the microbubble to the bulk of liquid medium can be expressed as
NA ¼ K L;A ðC Ag LA fC AL;1 Þ
ð18Þ
where CAg is molar concentration of gas A inside the microbubble.
ð9Þ C Ag ¼
where H’ is Henry’s constant, kG is local gas phase mass transfer coefficient, hA is shell permeability, kL is local liquid mass transfer coefficient. Assuming that the resistance in gas phase is negligible as compared to shell and liquid phase, Eq. (9) becomes,
1 1 1 ¼ þ K L hA kL
2.2.2. Dissolution of a microbubble in an aqueous medium containing dissolved gas other than the gas in microbubble core: Two way mass transfer Fig. 2 presents a schematic of a microbubble and the concentration profiles inside and close vicinity of microbubble dissolving in a multigas environment. It is assumed that the microbubble comprises of gas A and it is exposed to an aqueous medium which contains gas B dissolved in it. Taking balance for A over a microbubble
NA ¼
2 2 2 2 dR 3K L LA fRR0 Patm þ 2ðr0 Es ÞR0 þ 2Es R g BTRR0 C AL;1 ¼ dt 3Patm RR20 þ 4ðr0 Es ÞR20 þ 8Es R2
1 1 1 1 ¼ þ þ K L H0 kG hA kL
ð16Þ
3nA 4pR3
ð19Þ
nA is moles of gas A in Microbubble, R is radius of microbubble. CAL,1 is the concentration of a dissolved gas A far away from bubble in
ð10Þ
Also, assuming that no convections occur in the storage medium, the local liquid phase mass transfer coefficient (kL) can be estimated assuming Sh = 2 as follows:
kL ¼
DAL R
ð11Þ
Overall mass transfer coefficient then can be written as
KL ¼
1 hA
1 þ DRAL
Substituting Eq. (12) in Eq. (8)
ð12Þ Fig. 2. Schematic of a microbubble dissolving in a multigas environment.
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bulk of the aqueous medium (mol/m3), KL,A is overall mass transfer coefficient of gas A based on liquid phase (m/s), LA is Ostwald coefficient of gas and f is the saturation level of a dissolved gas in the surrounding medium. Substituting Eqs. (19) in (18).
NA ¼ K L;A
3nA LA 3
4p R
fC AL;1
ð20Þ
Comparing Eqs. (20) and (17)
dnA 3nA LA ¼ 4pR2 K L;A fC AL;1 dt 4pR3
ð21Þ
Overall mass transfer coefficient can be estimated as
K L;A ¼
1
The molar flux of B can be written as
NB ¼ K L;B ðfC BL;1 C Bg LB Þ
where CBg is molar concentration of gas A inside the microbubble. B C Bg ¼ 43n and nB is moles of gas B in microbubble, R is radius of pR3 microbubble, CBL,1 is a concentration of dissolved gas B in the bulk of aqueous medium (mol/m3), KL,B is overall mass transfer coefficient of gas B based on liquid phase (m/s), LB is Ostwald coefficient of gas (unitless) and f is the degree of saturation of the surrounding medium with respect to the dissolved gas. Comparing Eqs. (26) and (27),
dnB 3nB LB ¼ 4pR2 K L;B fC BL;1 dt 4pR3
K L;B ¼
where OS,A is mass transfer resistance to gas through microbubble shell. Substitute Eq. (22) in (21),
dnA 4pR2 3nA LA ¼ fC AL;1 dt Xs;A þ DRAL 4pR3
ð23Þ
1
ð29Þ
Xs;B þ DRBL
Substituting Eq. (29) in (28),
dnB 1 ¼ 4pR2 C Bg LB fC BL;1 dt Xs;B þ DR
pr2p ðr00 rÞ
ð30Þ
BL
Using energy theory for shell resistance
!
using energy barrier theory [8] for shell resistance,
Xs;A ¼ Xn;A exp
ð28Þ
Overall mass transfer coefficient can be written as
ð22Þ
Xs;A þ DRAL
ð27Þ
!
pr2p ðr00 rÞ
Xs;B ¼ Xn;B exp
ð31Þ
kb T
ð24Þ
kb T
Substituting Eq. (31) in (30), Substituting Eq. (24) in (23) 2
dnA 4pR 3nA LA pr2 ðr0 rÞ ¼ fC AL;1 3 p dt R 0 Xn;A exp þ DAL 4pR k T
ð26Þ
Eqs. (25) and (32) yield variation in number of moles of gas A and gas Bin the microbubble. To find out variation in microbubble radius with time overall balance can be written as
b
Similarly taking balances for gas B over a microbubble,
NB ¼
1 dnB 4pR2 dt 2
ð25Þ
where NB is mole flux of gas B mol/m /s, R is radius of microbubble, nB is moles of gas B in microbubble,
n;B
exp
1 pr2 ðr0 rÞ
3nB LB
dnB ¼ 4pR2 dt X
p
0
kb T
þ DRBL
4pR3
fC BL;1
ð32Þ
2r 4 3 Patm þ pR ¼ ðnA þ nB ÞBT 3 R
ð33Þ
Table 1 The estimated values of shell parameters and comparison of predicted values of shell resistances and the dissolution ties with the values reported in the literature. In the case of data presented below, the microbubble is dissolving in an aqueous medium containing dissolved gas same as the gas in the core of the microbubble (one way mass transfer). System
SDS coated air microbubble dissolving in air saturated water (f = 1) [9] SDS coated SF6 microbubble dissolving in an SF6 saturated water (f = 1) [9] Di12PC coated air microbubble dissolving in air saturated water (f = 0.833) [4] Di16PC coated air microbubble dissolving in air saturated water (f = 0.833) [4] Di18PC coated air microbubble dissolving in air saturated water (f = 0.833) [4] Di20PC coated air microbubble dissolving in air saturated water (f = 0.833) [4] Di22PC coated air microbubble dissolving in air saturated water (f = 0.833) [4] DMPC coated air microbubble dissolving in air saturated water (f = 0.833) [9] DPPC coated air microbubble dissolving in air saturated water (f = 0.833) [9] DSPC coated air microbubble dissolving in air saturated water (f = 0.833) [9] DAPC coated air microbubble dissolving in air saturated water (f = 0.833) [9] SDS-DPPC coated air microbubble dissolving in air saturated water (f = 1) [9] a
Es 105 (N/m) On 103 (s/m) r0 (N/m) Shell resistance (s/m)
tdiss (s)
Experimental Predicteda
Experimental Predicted % ARD in tdiss 50
1.673
1.01
0.0390
0
1018.27–1018.27
53.3
6.60
1.670
6.095
0.0390
0
26406.92–26426.52 386.4
499.2
29.19
120
0.4
0.0246
0
1290.68–1329.20
32.8
32.4
1.22
481
1.4
0.0249
5100 ± 2700
4484.02–5045.12
45.3
43.8
3.31
800
3.1
0.0247
8600 ± 5600
9978.11–12159.47
80.8
81.6
0.99
1000
6.2
0.0248
16,600 ± 5500 19906.96–31135.56 141
148
4.96
1810
6.8
0.0249
31,500 ± 7200 21779.55–27884.04 182
185
1.65
123
6.434
0.0246
13,070
20517.25–21144.04 103
94.8
7.96
410
8.0106
0.0249
14,664
25358.04–28033.62 129
120.1
6.90
900
15.02
0.0257
16,227
46625.15–58109.59 235
221.7
5.66
1030
17.01
0.0249
18,081
53846.18–69277.99 286
264.4
7.55
1.032
0.55
0.0283
0
1602.09–1602.50
79.7
3.24
77.2
A range in shell resistance (lowest and highest value) is given as the shell resistance increases during the microbubble dissolution.
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S.V. Dalvi, J.R. Joshi / Journal of Colloid and Interface Science 437 (2015) 259–269 Table 2 List of physical properties used in calculations. Physical property Coefficient of diffusivity of air in water Coefficient of diffusivity of SF6 in water Ostwald coefficient of air Ostwald coefficient of SF6 Temperature Collision diameter of air Collision diameter of SF6
3. Methodology of computation Value 9
2.05 10 m/s 1.2 109 m/s 1.71 102 5.4 103 295 K 3.58 1010 m 4.8 1010 m
3.1. Regression of shell properties from experimental data for microbubble dissolution A polynomial trend line was fitted to the experimental microbubble dissolution data (of Radius vs time) available in the literature [4,9]. Change in radius with time dR was then calculated dt by differentiating the polynomial equation with respect to time. To estimate the values of Es, On and r0, dR was calculated at t = 0, t = t dt and t = tdiss. This results into three nonlinear equations as shown below: At t = tdiss, R = 0, hence,
dR 3LA ¼ dt 2X exp pr2p Es n k T
ð35Þ
b
At t = 0, R = R0, hence,
3 2 2 3 dR 3 LA fR0 Patm þ 2ðr0 Es ÞR0 þ 2Es R0 g BTR0 C AL;1 ¼ dt Xn þ R0 3Patm R3 þ 4ðr0 Es ÞR2 þ 8Es R2 0
DAL
0
ð36Þ
0
At t = t, R = R, hence,
Fig. 3. Variation in dimensionless radius (R/R0) during dissolution of Di18PC coated air microbubble dissolving in water containing air dissolved at different saturation conditions (r0 = 0.0247 N/m, On = 3100 s/m, Es = 8 103 N/m).
3 LA fRR20 P atm þ 2ðr0 Es ÞR20 þ 2Es R2 g BTRR20 C AL;1 dR ¼ dt pr2p ðr00 rÞ R þ 3Patm RR20 þ 4ðr0 Es ÞR20 þ 8Es R2 Xn exp k T DAL b
ð37Þ The above three equations [Eqs. (35)–(37)] contain only three unknowns: Es, Xn and r0. The equations were solved simultaneously and the values of Es, Xn and r0 were estimated. 3.2. Estimating the dissolution kinetics for one way mass transfer i. Surface tension at the gas–liquid interface at any instant was calculated using Eq. (4) and pressure inside the microbubble was calculated using Eq. (3). ii. Shell resistance of gas was calculated by Eq. (15) and then overall mass transfer coefficient was estimated by Eq. (12). iii. Radius of the microbubble at every instant was then calculated using Eq. (16). 3.3. Estimating the dissolution kinetics for two way mass transfer i. Surface tension at the gas–liquid interface at any instant was calculated using Eq. (4) and pressure inside the microbubble was calculated using Eq. (3). ii. Shell resistance to gas diffusion was calculated using Eqs. (24) and (31) and then overall mass transfer coefficients were estimated using Eqs. (22) and (29). iii. The variation in moles of A at any instant was calculated as For gas A: 2
Fig. 4. Variation in (a) surface tension and (b) shell resistance with time at different saturation conditions during the dissolution of Di18PC coated air microbubble in air saturated water.
sþ1
nA
4pðRs Þ 3nsA LA ¼ nA Dt s fC AL;1 Rs XS;A þ DAL 4pðRs Þ3
!
s
ð38Þ
For gas B 2
The above equation after rearrangement becomes
4pP atm 3 8pr 2 R þ R ðnA þ nB Þ ¼ 0 3BT 3BT Eq. (34) can be solved to obtain values of R with time.
nBsþ1 ¼ nsB Dt ð34Þ
4pðRs Þ 3nsB LB fC BL;1 Rs XS;B þ DBL 4pðRs Þ3
! ð39Þ
iv. A microbubble radius at any instant was then estimated by solving the following equation.
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Table 3 Variation in dissolution time (tdiss), shell resistance and surface tension at the gas–liquid interface (coated with shell material) during dissolution of microbubbles filled with different gases and encapsulated in different shell materials in an aqueous medium containing varying degree of saturation of a dissolved gas. In the case of data presented below, the microbubble is dissolving in an aqueous medium containing dissolved gas same as the gas in the core of the microbubble (one way mass transfer). System
R0 (lm)
f
tdiss (s)
Range of XS,A (s/m) variation
Range of r (N/m) variation
SDS coated air microbubble dissolving in air saturated water
5
0 0.2 0.4 0.6 0.8 1
0.5 0.6 0.8 1.1 1.6 2.9
1018.27–1018.27 1018.27–1018.27 1018.27–1018.27 1018.27–1018.27 1018.27–1018.27 1018.27–1018.27
0.039–0.03898 0.039–0.03898 0.039–0.03898 0.039–0.03898 0.039–0.03898 0.039–0.03898
Di12PC coated air microbubble dissolving in air saturated water
5
0 0.2 0.4 0.6 0.8 1
0.7 0.8 0.1 1.2 1.6 2.4
1290.68–1329.20 1290.68–1328.80 1290.68–1329.40 1290.68–1329.03 1290.68–1329.16 1290.68–1329.20
0.0246–0.02341 0.0246–0.02342 0.0246–0.02340 0.0246–0.02341 0.0246–0.02341 0.0246–0.0234
Di16PC coated air microbubble dissolving in air saturated water
5
0 0.2 0.4 0.6 0.8 1
1.4 1.6 2 2.6 3.6 5.4
4484.02–5045.12 4484.02–5040.18 4484.02–5043.53 4484.02–5046.67 4484.02–5046.67 4484.02–5049.35
0.0249–0.02013 0.0249–0.02017 0.0249–0.02014 0.0249–0.02012 0.0249–0.0201 0.0249–0.02011
Di18PC coated air microbubble dissolving in air saturated water
5
0 0.2 0.4 0.6 0.8 1
3.4 4 4.75 5.9 7.85 11.8
9978.11–12159.47 9978.11–12158.34 9978.11–12159.03 9978.11–12159.48 9978.11–12159.02 9978.11–12159.31
0.0247–0.0167 0.0247–0.0167 0.0247–0.0167 0.0247–0.0167 0.0247–0.0167 0.0247–0.0167
Di20PC coated air microbubble dissolving in air saturated water
5
0 0.2 0.4 0.6 0.8 1
7.2 8.4 10 12.6 16.8 25.6
19906.96–31135.56 19906.96–31136.13 19906.96–31136.83 19906.96–31136.21 19906.96–31136.57 19906.96–31136.82
0.0248–0.0067 0.0248–0.0067 0.0248–0.0067 0.0248–0.0067 0.0248–0.0067 0.0248–0.0067
Di22PC coated air microbubble dissolving in air saturated water
5
0 0.2 0.4 0.6 0.8 1
7.9 9.3 11.3 14.3 19.6 31.3
21779.55–27884.04 21779.55–27884.01 21779.55–27881.04 21779.55–27884.32 21779.55–27883.83 21779.55–27885.55
0.0249–0.0149 0.0249–0.0149 0.0249–0.01491 0.0249–0.0149 0.0249–0.0149 0.0249–0.0149
SDS coated SF6 microbubble dissolving in an SF6 saturated water
5
0 0.2 0.4 0.6 0.8 1
23.4 27.4 33.2 42.4 59.2 105.2
26406.92–26426.52 26406.92–26426.52 26406.92–26426.52 26406.92–26426.52 26406.92–26426.52 26406.92–26426.52
0.039–0.03898 0.039–0.03898 0.039–0.03898 0.039–0.03898 0.039–0.03898 0.039–0.03898
DMPC PEG2000S coated air microbubble dissolving in air saturated water
5
0 0.2 0.4 0.6 0.8 1
5.9 7.0 8.7 11.7 17.8 42.0
20517.25–21144.04 20517.25–21143.91 20517.25–21143.79 20517.25–21144.05 20517.25–21143.98 20517.25–21143.97
0.0246–0.02337 0.0246–0.02337 0.0246–0.02337 0.0246–0.02337 0.0246–0.02337 0.0246–0.02337
DPPC PEG2000S coated air microbubble dissolving in air saturated water
5
0 0.2 0.4 0.6 0.8 1
7.71 9.20 11.44 15.26 23.35 55.86
25358.04–28033.62 25358.04–28033.62 25358.04–28033.62 25358.04–28033.62 25358.04–28033.62 25358.04–28033.62
0.0249–0.0208 0.0249–0.0208 0.0249–0.0208 0.0249–0.0208 0.0249–0.0208 0.0249–0.0208
DSPC PEG2000S coated air microbubble dissolving in air saturated water
5
0 0.2 0.4 0.6 0.8 1
15.10 18.11 22.67 30.27 46.32 108.24
46625.15–58109.59 46625.15–58109.59 46625.15–58109.59 46625.15–58109.59 46625.15–58109.59 46625.15–58109.59
0.0257–0.0167 0.0257–0.0167 0.0257–0.0167 0.0257–0.0167 0.0257–0.0167 0.0257–0.0167
DAPC PEG2000S coated air microbubble dissolving in air saturated water
5
0 0.2 0.4 0.6 0.8 1
18.04 21.60 27.02 36.29 56.18 136.57
53846.18–69277.99 53846.18–69277.99 53846.18–69277.99 53846.18–69277.99 53846.18–69277.99 53846.18–69277.99
0.0249–0.0146 0.0249–0.0146 0.0249–0.0146 0.0249–0.0146 0.0249–0.0146 0.0249–0.0146 (continued on next page)
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S.V. Dalvi, J.R. Joshi / Journal of Colloid and Interface Science 437 (2015) 259–269 Table 3 (continued) System
R0 (lm)
f
tdiss (s)
Range of XS,A (s/m) variation
Range of r (N/m) variation
SDS-DPPC coated air microbubble dissolving in air saturated water
5
0 0.2 0.4 0.6 0.8 1
0.76 0.90 1.12 1.48 2.22 4.78
1602.09–1602.50 1602.09–1602.50 1602.09–1602.50 1602.09–1602.50 1602.09–1602.50 1602.09–1602.50
0.0283–0.02829 0.0283–0.02829 0.0283–0.02829 0.0283–0.02829 0.0283–0.02829 0.0283–0.02829
4pP atm ðsþ1Þ 3 8prðsÞ ðsþ1Þ 2 ðsþ1Þ ðsþ1Þ þ ðR Þ nA þ nB ¼0 R 3BT 3BT
ð40Þ
The time step (Dt) was varied as,
Dt ¼
R2 4Dmin
ð41Þ
where Dmin is the diffusivity of a gas with lowest diffusivity. 4. Results and discussions 4.1. Estimation of shell properties
Fig. 5. Variation in R/R0 for dissolution of SDS coated SF6 microbubble in water saturated with air at different saturation levels (r0 = 0.039 N/m, On = 1010 s/m, Es = 1.673 105 N/m).
7.00E-08
Molar Flux (kmol/m2 s)
The values of shell properties such as On, Es and r0 were estimated as discussed in Section 3.1. The experimental data on microbubble dissolution available in the literature [4,9] were used. The estimated values of On, Es and r0 are reported in Table 1. It can be observed from Table 1 that the elasticity of SDS as a shell material is much lower (at least by 2 orders of magnitude) than the elasticity of lipids (Di12PC to Di22PC). Therefore, SDS can be considered to be relatively inelastic as compared to lipids. Also, in case of lipids, the shell elasticity and shell resistance increases as the length of hydrophobic chain increases from Di12PC to Di22PC lipids. Table 1 presents a comparison of estimated shell resistances with the values available in the literature [4,9]. Comparison shows that the shell resistances reported in literature are lower than the values estimated in this work for most of the cases. The values reported in the literature were obtained without including shell elasticity or the variation in shell resistances with time which could be the reason for this disparity. The values of shell parameters thus estimated were further used for prediction of dissolution kinetics of microbubbles.
5.00E-08
NB 3.00E-08
1.00E-08
-1.00E-08
0.1
NA
1
10
100
1000
Time (seconds)
4.2. Kinetics of microbubble dissolution in an aqueous medium saturated/partially saturated with a gas used to make microbubble: One way mass transfer
Fig. 6. Variation in molar flux of SF6 (NA) and that of air (NB) with time during dissolution of SDS coated SF6 microbubble in water saturated with air (f = 1). Positive values of NB indicate influx of air into the microbubble whereas negative values of NA indicate efflux of SF6 from the microbubble.
Eq. (16) was solved numerically with initial condition of R = R0 at t = 0 to obtain variation in the radius of a microbubble dissolving in an aqueous medium. The values of On, Es and r0 presented in Table 1, were used along with the constants listed in Table 2 during the calculations. These calculations were performed in order to validate the proposed model and verify if the estimated values of shell properties can correctly predict the dynamics of microbubble dissolution. The predicted dissolution data match very well with the experimental dissolution data with R2 values ranging from 0.94 to 0.98 for all the systems studied in this work (See Figs. S1–S3). The values of dissolution times obtained also match closely with the experimental values of dissolution times for all the systems studied (Table 1). To explain the dynamics of the microbubble dissolution, a case of dissolution of Di18PC coated air microbubble in air saturated aqueous medium has been chosen. Fig. 3 presents predicted variation in R/R0 with time (solid lines) for Di18PC coated air bubble during its dissolution in an aqueous medium containing
air at different saturation levels. Experimental dissolution data for dissolution of a microbubble in an aqueous medium with air saturation level (f) of 0.833 (filled circles) are also presented along with the calculated dissolution profiles. It can be observed that the experimental data (for f = 0.833) matche very well with the predicted data (R2 = 0.98). Further, variation in surface tension (r) at the lipid coated air–water interface with time (Fig. 4a), and variation in lipid shell resistance (OSA) with time (Fig. 4b) has also been estimated. It was found that surface tension (r) decreases almost linearly with time (Fig. 4a) whereas the shell resistance increases with time (Fig. 4b). As microbubble dissolves, molecules of shell material get closer and get fully compressed which makes the microbubble shell increasingly rigid. This results in a decrease in surface tension with time. The compression of the molecules of shell material, on the other hand increases resistance for mass transfer of a gas due to a decrease in pore sizes in the compressed shell.
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Table 4 Variation in dissolution time (tdiss), shell resistance and surface tension at the gas–liquid interface (coated with shell material) during dissolution of microbubbles filled with SF6 and encapsulated in different shell materials in an aqueous medium containing varying degree of saturation of a dissolved air. In the case of data presented below, the microbubble dissolves in an aqueous medium containing a dissolved gas other than the core gas of the microbubble (two way mass transfer). Shell material
f
Variation in XS,A (s/m)
Variation in XS,B (s/m)
Variation in r (N/m)
SDS
0 0.2 0.4 0.6 0.8 1
6.73 7.27 8.02 9.19 11.46 21.18
6095–6099.48 6095–6099.48 6095–6099.48 6095–6099.48 6095–6099.48 6095–6099.48
1010–1010.41 1010–1010.41 1010–1010.41 1010–1010.41 1010–1010.41 1010–1010.41
0.039–0.03898 0.039–0.03898 0.039–0.03898 0.039–0.03898 0.039–0.03898 0.039–0.03898
Di12PC
0 0.2 0.4 0.6 0.8 1
2.69 3.03 3.52 4.34 6.17 16.86
971.50–1024.15 971.50–1024.15 971.50–1024.15 971.50–1024.15 971.50–1024.15 971.50–1024.15
400–411.92 400–411.92 400–411.92 400–411.92 400–411.92 400–411.92
0.0247–0.0235 0.0247–0.0235 0.0247–0.0235 0.0247–0.0235 0.0247–0.0235 0.0247–0.0235
Di16PC
0 0.2 0.4 0.6 0.8 1
5.18 5.66 6.34 7.43 9.72 21.69
3387.53–4185.40 3387.52–4185.41 3387.52–4185.42 3387.52–4185.432 3387.52–4185.45 3387.52–4185.55
1400–1574.794 1400–1574.796 1400–1574.798 1400–1574.8 1400–1574.804 1400–1574.825
0.0249–0.02009 0.0249–0.02009 0.0249–0.02009 0.0249–0.02009 0.0249–0.02009 0.0249–0.02009
Di18PC
0 0.2 0.4 0.6 0.8 1
10.03 10.74 11.70 13.17 16.02 30.32
7529.14–10704.18 7529.14–10704.18 7529.14–10704.18 7529.147–10704.18 7529.147–10704.18 7529.147–10704.18
3100–3770.192 3100–3770.192 3100–3770.192 3100–3770.192 3100–3770.193 3100–3770.193
0.0247–0.0167 0.0247–0.0167 0.0247–0.0167 0.0247–0.0167 0.0247–0.0167 0.0247–0.0167
Di20PC
0 0.2 0.4 0.6 0.8 1
21.08 22.31 23.95 26.32 30.56 50.61
16453.7–25542.76 16453.7–25542.77 16453.7–25542.79 16453.7–25542.8 16453.7–25542.82 16453.7–25542.91
6200–9653.897 6200–9653.903 6200–9653.911 6200–9653.919 6200–9653.931 6200–9653.987
0.0247–0.0066 0.0247–0.0066 0.0247–0.0066 0.0247–0.0066 0.0247–0.0066 0.0247–0.0066
Di22PC
0 0.2 0.4 0.6 0.8 1
25.21 26.41 27.93 30.02 33.52 52.17
15058.29–33380.47 15058.29–33380.51 15058.29–33380.55 15058.29–33380.61 15058.29–33380.68 15058.29–33381.03
6800–8684.771 6800–8684.773 6800–8684.775 6800–8684.778 6800–8684.782 6800–8684.798
0.0249–0.0149 0.0249–0.0149 0.0249–0.0149 0.0249–0.0149 0.0249–0.0149 0.0249–0.0149
DMPC PEG2000
0 0.2 0.4 0.6 0.8 1
15.88 17.12 18.85 21.56 27.03 58.34
16226.75–17137.29 16226.75–17137.32 16226.75–17137.36 16226.75–17137.43 16226.75–17137.52 16226.75–17137.99
6434–6632.40 6434–6632.40 6434–6632.41 6434–6632.43 6434–6632.45 6434–6632.55
0.0246–0.02337 0.0246–0.02337 0.0246–0.02337 0.0246–0.02337 0.0246–0.02337 0.0246–0.02337
DPPC PEG2000
0 0.2 0.4 0.6 0.8 1
21.37 22.83 24.82 27.84 33.58 62.38
20085.05–24431.41 20085.05–24431.44 20085.05–24431.49 20085.05–24431.54 20085.05–24431.61 20085.05–24431.94
8010.60–8932.85 8010.60–8932.85 8010.60–8932.86 8010.60–8932.87 8010.60–8932.89 8010.60–8932.95
0.0249–0.02049 0.0249–0.02049 0.0249–0.02049 0.0249–0.02049 0.0249–0.02049 0.0249–0.02049
DSPC PEG2000
0 0.2 0.4 0.6 0.8 1
43.02 45.52 48.78 53.41 61.42 96.40
37076.36–55279.11 37076.36–55280.11 37076.36–55281.44 37076.36–55283.27 37076.36–55286.13 37076.36–55297.27
15,020–18756.95 15,020–18757.14 15,020–18757.39 15,020–18757.74 15,020–18758.28 15,020–18760.38
0.0257–0.0167 0.0257–0.0167 0.0257–0.0167 0.0257–0.0167 0.0257–0.0167 0.0257–0.0167
DAPC PEG2000
0 0.2 0.4 0.6 0.8 1
52.56 55.36 58.97 64.04 72.59 110.44
42649.32–67397.25 42649.32–67397.32 42649.32–67397.39 42649.32–67397.48 42649.32–67397.59 42649.32–67398.00
17,010–21940.75 17,010–21940.76 17,010–21940.77 17,010–21940.79 17,010–21940.81 17,010–21940.88
0.0249–0.0146 0.0249–0.0146 0.0249–0.0146 0.0249–0.0146 0.0249–0.0146 0.0249–0.0146
tdiss (s)
Another interesting observation that could be made from Fig. 4a and b is that the rate of change of surface tension and shell resistance was found to be higher at the lower saturation levels as compared to the higher saturation levels of the aqueous medium. At lower saturation levels of the gas in the surrounding medium, the concentration gradient for the efflux of a gas from the core of the gas increases. This causes faster dissolution of a microbubble
leading to faster shrinkage/compression of shell material and hence higher rate of change of surface tension and the shell resistance. Table 3 presents variation in shell resistance and partial pressure of a gas in a microbubble during its dissolution for microbubbles made of different shell materials and different gases. It can be observed that for a material like SDS, the variation in shell
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Fig. 7. Variation in R/R0, and partial pressure of SF6 and that of air with time for different saturation conditions, for dissolution of Di18PC coated SF6 microbubble in water saturated with air (r0 = 0.0247 N/m, On = 3100 s/m, Es = 8 103 N/m).
resistance and surface tension with time is negligible (Table 3). However, the variation in surface tension and shell resistance to diffusion of gas is higher for lipids. This could be attributed to the higher elasticity of the lipid shells as compared to shell made of SDS.
4.3. Kinetics of microbubble dissolving in an aqueous medium containing dissolved gas other than the gas used to make the microbubble: Two way mass transfer The estimated values of shell parameters were further used for prediction of dissolution kinetics of SF6 filled microbubbles (R0 = 25 lm) coated with materials such as lipids (Di12PC to Di22C) and SDS in an aqueous medium containing air dissolved in it. In order to validate the proposed model for prediction of microbubble dissolution dynamics in a multigas environment, the predicted dissolution data for dissolution of SF6 filled SDS coated microbubble are compared with the dissolution data available in the literature [10]. Fig. 5 presents variation in non-dimensional radius (R/R0) with time for dissolution of SF6 filled SDS coated microbubble in water containing air dissolved at different saturation levels (f). The data predicted for dissolution of SF6 filled SDS coated microbubble in air saturated (f = 1) water were compared with the experimental dissolution data obtained from literature [10], as shown in Fig. 5. It can be observed from Fig. 5 that the model correctly captures dissolution kinetics of a microbubble. A microbubble growth phase is observed initially as a result of higher influx of air (NB) into the microbubble as compared to the efflux of SF6 (NA) out of the microbubble (Fig. 6). However, the mass transfer of air to the microbubble core (NB) increases the concentration of air in the microbubble core. This reduces the concentration gradient for the mass transfer of air from the surroundings to the microbubble core. Hence the flux of air (NB) decreases continuously with time during the microbubble dissolution (Fig. 6). A continuous decrease in influx of air (NB) along with the efflux of SF6 (NA) slows down the rate of increase in microbubble size during the growth phase. Eventually, the shrinking phase of a microbubble is triggered when the influx of air (NB) reduces to a very low value and even reverses the direction (shown in Fig. 6). The reversal in the direction of the flux of air (negative flux) indicates that the air now starts dissolving in the surrounding medium. This reversal happens as the reduction in microbubble size results in an increase in Laplace pressure inside the microbubble. Increase in Laplace pressure generates a concentration gradient for the transfer of air from
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the microbubble core to the surroundings and hence air starts dissolving in the surrounding medium. A continuous decrease in the number of moles from the microbubble core due to the effluxes of SF6 as well as air, results in a decrease in the microbubble size to almost zero. It can also be observed from Fig. 5 that the microbubble dissolution behavior changes drastically with a decrease in saturation level (f) of the aqueous medium. At higher saturation levels such as f = 1, a microbubble dissolution exhibits a growth phase followed by a shrinking phase. However, as the value of f decreases (f < 1), the growth phase disappears and the microbubble only shrinks during the dissolution process. This is mainly because of a decreased amount of dissolved air in the aqueous medium with a decrease in f. A decrease in dissolved air in the aqueous medium reduces the influx of air to the microbubble and hence a gradual disappearance of the growth phase. In fact, a microbubble growth phase is not observed at lower values of f (such as f = 0). It can also be observed from Fig. 5 that the dissolution time (tdiss) of a microbubble increases as the saturation level (f) of the medium increases. An increase in the saturation level, increases the concentration of dissolved air in aqueous medium. Increase in the amount of dissolved air in the aqueous medium causes microbubble to grow, which otherwise would have only shrunk in the absence of any dissolved air in the surrounding medium (f = 0). The existence of additional growth phase along with shrinking phase increases the dissolution time of the microbubble in the surrounding medium. Table 4 presents predicted dissolution time, variation in shell resistances and surface tensions for dissolution of SF6 filled microbubbles made of various shell materials in an aqueous medium containing air at different saturation levels ( f). It can be observed that for SDS coated SF6 microbubble, surface tension and the shell resistances to the mass transfer of air and SF6 do not vary significantly during the process of microbubble dissolution. This could be attributed to a lower value of elasticity of SDS shell. However, in case of lipids, where elasticity values are higher than that for SDS, a significant variation in the shell resistance to gases and surface tension during microbubble dissolution has been observed (Table 4). Therefore, to illustrate the variation in shell resistances and surface tension during the microbubble dissolution, a case of dissolution of Di18PC coated SF6 microbubble in air saturated aqueous medium has been chosen. Fig. 7 presents the variation of dimensionless radius (R/R0) with time for different saturation conditions during dissolution of Di18PC coated SF6 microbubble in water containing dissolved air. As was observed for the dissolution of SDS coated SF6 microbubble, the time of dissolution was found to increase with increase in air saturation level of water for Di18PC microbubble as well. Fig. 8(a), (b) and (c) present variation in surface tensions and shell resistance with time during the process of microbubble dissolution. The surface tension (r) at the gas–liquid interface exhibits a maximum whereas the shell resistances to the mass transfer of SF6 (OSA) and air (OSB) exhibit minimum during the process of microbubble dissolution. During the growth phase, as microbubble grows, molecules of the shell material move away from each other and get decompressed. This reduces the rigidity of the shell resulting in a decrease in interaction among molecules of the shell and hence surface tension increases. During the shrinking phase however, as microbubble dissolves, the molecules of the shell material get closer and get fully compressed. The shell becomes increasingly rigid and hence surface tension decreases with time. On the shell resistances however, the decompression and compression of shell material during the growth and shrinking phase respectively has an exactly opposite effect. As the microbubble grows, the gap between the molecules of the shell material increases with time. This increases gas perme-
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Fig. 8. (a) Variation in surface tension with time, and (b and c) variation in shell resistance for SF6 and air with time at different saturation conditions for Di18PC coated air microbubble dissolving in air saturated water.
ability through shell material and hence shell resistance decreases. However, the molecules of shell material come closer as microbubble shrinks, resulting in a tighter packing and increased interaction among the shell molecules which reduces the defects in the shell structure and restricts the diffusion of gas molecules through the shell. This results in an increase in the resistance for the mass transfer of gas. 5. Conclusions A new model for the prediction of dissolution kinetics of a microbubble in an aqueous environment containing dissolved gases is presented. The process of microbubble dissolution and the dissolution time both are greatly affected by the shell properties such as surface tension (r) at the gas–liquid interface stabilized by the shell molecules, mass transfer resistance (X) offered by the shell material and the shell elasticity (Es). However, the reports available in the literature on the modeling of microbubble dissolution [2–7,10] present models which do not take into account the influence of above mentioned shell properties and their variation on microbubble dissolution process. Moreover, the values of the above mentioned shell parameters are not known/ available and hence arbitrary values for these variables have been assumed in many of the reports [2,3,5–7,10] for estimation of dissolution kinetics. Therefore, a new model has been presented in this work, which takes into account the effect of shell parameters and their variation on the microbubble dissolution. The values of these shell properties have also been estimated using the proposed model and the experimental dissolution data available in the literature [4,9]. The results show that the lipid shell is more elastic than
the shell made of SDS as elasticity of a SDS shell was found to be much lower (at least by 2 orders of magnitude) than the elasticity of lipids (Di12PC to Di22PC). Comparison of estimated shell resistances with the values reported in the literature [4,9] shows that the shell resistances reported in literature are lower than the values estimated in this work for most of the cases. The estimated values of shell parameters were further used for prediction of dissolution kinetics of microbubbles in an aqueous medium containing dissolved gases. The predicted dissolution data match very well with the experimental data with R2 values ranging from 0.94 to 0.98 for all the systems studied in this work. Further analysis of the results shows that the surface tension (r) decreases almost linearly with time and the shell resistance increases with time during the microbubble dissolution in case of a one way mass transfer of a gas (i.e. only from microbubble core to the surroundings). However, during the microbubble dissolution in a multigas environment where a two way mass transfer occurs (i.e. from microbubble core to the surroundings and from surrounding medium to the microbubble core), the surface tension at the gas–liquid interface exhibits a maximum whereas the shell resistances to the mass transfer of gas (OSA) and air (OSB) exhibit minimum. It was also found that the variation in surface tension and shell resistances during the dissolution process is highly dependent on the shell elasticity. The lower shell elasticity was found to result in a negligible variation in the shell resistance and surface tension (as observed in the case of SDS) whereas higher shell elasticity was found to result in a large variation in surface tension and shell resistance during the microbubble dissolution (as observed in case of lipids). This is in contrast to the reports available in the literature [2–7,10] where surface tension and shell resistance was assumed
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to be constant and no dependence of these shell properties on shell elasticity was taken into account. The model presented in this work, can further be extended to predict the circulation persistence time of microbubbles injected into a blood stream and to predict the shelf stability of any microbubbles formulation. Moreover, the methodology presented in this work can be used for the purpose of designing a microbubble system with desired characteristics and performance. Acknowledgments Authors gratefully acknowledge the financial support from the Department of Biotechnology of Ministry of Science and Technology, and Ministry of Human Resource Development of Government of India, and Indian Institute of Technology, Gandhinagar (IITGN).
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Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.jcis.2014.09.044. References [1] [2] [3] [4] [5] [6] [7]
S. Sirsi, M. Borden, Bubble Sci. Eng. Technol. 1 (1–2) (2009) 3–17. S.P. Epstein, J. Chem. Phys. 18 (11) (1950) 1505–1509. P.B. Duncan, D. Needham, Langmuir 20 (7) (2004) 2567–2578. M.A. Borden, M.L. Longo, Langmuir 18 (24) (2002) 9225–9233. K. Sarkar, A. Katiyar, P. Jain, Ultrasound Med. Biol. 35 (8) (2009) 1385–1396. A. Katiyar, K. Sarkar, P. Jain, J. Colloid Interface Sci. 336 (2) (2009) 519–525. M. Azmin, G. Mohamedi, M. Edirisinghe, E.P. Stride, Mater. Sci. Eng. C 32 (8) (2012) 2654–2658. [8] J.J. Kwan, M.A. Borden, Adv. Colloid Interface Sci. 183–184 (2012) 82–99. [9] M.M. Lozano, M.L. Longo, Langmuir 25 (6) (2009) 3705–3712. [10] J.J. Kwan, M.A. Borden, Langmuir 26 (9) (2010) 6542–6548.
