6420
J. Phys. Chem. B 2008, 112, 6420–6425
Effective Method of Characterizing Specific Liquid Fluorocarbon Interactions Using Ultrasound S. Ravi,*,† J. Amoros,‡ and K. Arockia Jayalatha§ Department of Physics, National College, Tiruchirapalli 620 001, India, Departmento de Fisica Applicada, UniVersitad de Cantabria, AVda, Los Castros, 39005 Santander, Spain, and Department of Physics, CauVery College for Women, Tiruchirapalli 620 018, India ReceiVed: January 17, 2008; ReVised Manuscript ReceiVed: February 24, 2008
Several studies have used the analytical results available for structure factor, osmotic pressure, vapor pressure, and suspension viscosity to characterize nanoparticle interactions. In this work a novel attempt has been made to characterize seven different types of liquid per fluorocarbon nanoparticles (LPFC-np) by estimating packing factor, segment diameter, chemical potential, Rao’s constant, and adiabatic and isothermal compressibilities using experimental ultrasonic velocity as input. The segment diameter has also been examined by using experimental surface tension and viscosity for comparison. The calculations were extended for different temperatures involving four different equations of state. We have tested the sensitivity of all these parameters to a very small change in the heat capacity ratio. This extensive calculation helps to make a reasonable description about the interactions among the LPFC-np. Also a better correlation could be determined between the interaction of ultrasound with LPFC-np (as a facilitator) and the pure absorption (propagation) of ultrasound by the entire system. 1. Introduction Ultrasound application for liquid per fluorocarbon nanoparticles (LPFC-np) holds interest because of their peculiar physical properties such as low surface tension and viscosity, their ability to transmit sound at lower speeds than any other liquids, and their string affinity to gases. The possibility of using liquids instead of air in the exchange of gases has always fascinated researchers. The most remarkable character of LPFC-np is their unexpectedly weak attractive interactions. Though there is little support for this interpretation,1–5 through this study it is made stronger. It is a well-known fact that the LPFC are being widely considered as nanoparticles while they are emulsified. But even in liquid form they happen to possess the same physical property as that of an LPFC emulsion. Many researchers have treated LPFC in liquid form as a nanoparticle that assists the delivery of drugs into the targeted cells. Literature from which the data has been taken for this study too labels liquid LPFC as a nanoparticle. Interactions between nanoparticles are characterized with the available analytical results for structure factor, osmotic pressure, vapor pressure, and suspension viscosity, to date.6–9 Also, studies were made on how ultrasound facilitates LPFC-np to deliver drug into the targeted cells.10–12 In this present study, we intend to characterize the interactions among the LPFC-np by using the experimental ultrasonic velocity measured at different temperatures by Marsh et al.,14 for the seven systems such as perfluorohexane (C6F14), perfluoroheptanes (C7F16), perfluoro1-butyltetrahyrofuran (C8F16O), perfluoro octanes (C8F18), perfluorodecalin (C10F18), perfluorodichloro octane (C8F16Cl2), and perfluorooctyl bromide (C8F17Br), as the only input. * Correspondening author. E-mail:
[email protected]. † National College. ‡ Universitad de Cantabria. § Cauvery College for Women.
Through this, we could estimate thermo-physical parameters like packing factor y, segment diameter d, chemical potential µ/kT, Rao’s constant R, adiabatic βa, and isothermal compressibility βT. The segment diameter d has also been examined by using experimental surface tension σ and viscosity η for comparison. The calculations were extended for different temperatures involving four different equations of state. We have tested the sensitivity of these parameters to a very small change in the heat capacity ratio γ. This extensive calculation has been made for all seven systems mentioned above. Our intention is to use the propagation of ultrasound through the system and interpret about the interactions between the LPFC-np. Also a reasonably successful correlation could be arrived between the interaction of ultrasound with LPFC-np (as a facilitator) and the pure absorption (propagation) of ultrasound by the entire system. For such calculations, we considered that the particles are modeled with a simple united atom approach: hard, spherical attractive segments of diameter d bounded together.13 It has been established that C10F18 has an edge over other systems because of its peculiar behavior.14,15 This is found true through this study too. Discussions have been carried out exclusively on this in later section. 2. Theoretical Model 2.1. Evaluation of Packing Factor y and Segment Diameter d. The basic aim of this work is to relate the experimental ultrasonic velocity U to an analytical equation of state, which solely depends on the packing factor y. Hence forth, from the X-ray diffraction studies on liquids, we have the equation for the static structure factor S(q) as16
S(q) ) 1 +
4πFn q
∫ [g(r) - 1] sin qr dr
(1)
where Fn is the number density and q the momentum transfer. At long wavelength limit, that is, q f 0, the above equation
10.1021/jp800812c CCC: $40.75 2008 American Chemical Society Published on Web 04/19/2008
Ultrasound To Characterize Fluorocarbon Interactions
J. Phys. Chem. B, Vol. 112, No. 20, 2008 6421
reduces to an analytical expression for the equation of state for liquids as
anl (y) ) XMCSL
S(0) ) FnkβT
(2)
Here, S(0) is nothing but the equation of state Hence, we could write anl
X
anl (y) ) XML
(E.O.S).17–19
(y) ) FnkTβT
-y5 + 4y4 - 9y3 + 8y2 + 8y + 2 (4) 2(1 - y)4
Obviously all four of the models exclusively depend on the packing factor y which is given by
(3)
Depending upon the interactions (existing in the liquid) that one could consider, the expression for Xanl(y) takes four different forms given by the Percus-Yevick virial20 approach (PY), scaled particle theory (SPT),21 Mansoori-Carnahan-StarlingLeland22 (MCSL), and Matyushov-Ladanyi (ML)23 as anl (y) ) XPY
y4 - 4y3 + 4y2 + 4y + 1 (1 - y)4
y)
π 6Fnd3
(5)
Now the right-hand side of eq 3 can be related to the ultrasonic velocity24 U through
(1 + 2y) (1 - y)2
FnkTβT )
(1 + 2y)2 anl (y) ) XSPT ( )3
1-y
RT γMU2
(6)
Hence forth from eqs 2–4 and 6 we have
TABLE 1: Packing Factor y and Diameter d PY (virial) temp, °C
SPT
MCSL
γ ) 1.1
γ ) 1.2
γ ) 1.1
γ ) 1.2
y
y
y
y
d (Å)
d (Å)
d (Å)
d (Å)
ML using σ, using η eq 9, d eq 10, d (Å) (Å) d (Å)
γ ) 1.1
γ ) 1.2
γ ) 1.1
γ ) 1.2
y
d (Å)
y
d (Å)
y
d (Å)
y
0.2425 0.2375 0.2331 0.2280 0.2230
5.367 5.347 5.330 5.307 5.285
0.2368 0.2323 0.2274 0.2228 0.2176
5.325 5.307 5.286 5.267 5.242
0.2425 0.2378 0.2331 0.2283 0.2233
5.367 5.349 5.330 5.310 5.287
5.322
5.324
5.263
5.265
25 30 35 40 45
0.4488 0.4415 0.4335 0.4255 0.4165
6.590 6.574 6.554 6.534 6.508
0.4578 0.4504 0.4425 0.4350 0.4265
6.633 6.618 6.599 6.582 6.560
0.2735 0.2682 0.2628 0.2571 0.2518
5.587 5.570 5.547 5.524 5.503
0.2800 0.2747 0.2693 0.2638 0.2579
5.631 5.612 5.593 5.572 5.547
1. C6F14 0.2365 5.323 0.2320 5.305 0.2275 5.287 0.2225 5.264 0.2175 5.241
25 30 35 40 45
0.4769 0.4695 0.4625 0.4555 0.4482
6.981 6.964 6.948 6.932 6.914
0.4856 0.4790 0.4720 0.4650 0.4575
7.023 7.010 6.995 6.980 6.962
0.2935 0.2885 0.2832 0.2785 0.2730
5.935 5.920 5.900 5.883 5.861
0.2999 0.2948 0.2900 0.2850 0.2795
5.981 5.963 5.947 5.929 5.907
2. C7F16 0.2538 5.657 0.2497 5.642 0.2450 5.622 0.2408 5.605 0.2360 5.583
0.2594 0.2552 0.2510 0.2462 0.2410
5.698 5.683 5.667 5.647 5.622
0.2540 0.2497 0.2452 0.2409 0.2364
5.659 5.642 5.623 5.606 5.586
0.2597 0.2553 0.2510 0.2465 0.2421
5.701 5.684 5.667 5.649 5.631
5.648
5.651
5.598
5.594
25 30 35 40 45
0.4957 0.4898 0.4831 0.4770 0.4700
7.191 7.181 7.166 7.154 7.137
0.5044 0.4984 0.4921 0.4855 0.4794
7.233 7.223 7.210 7.196 7.184
0.3070 0.3026 0.2980 0.2935 0.2886
6.130 6.116 6.100 6.084 6.066
0.3136 0.3089 0.3045 0.2999 0.2948
6.174 6.158 6.144 6.128 6.109
3. C8F16O 0.2658 5.842 0.2618 5.828 0.2579 5.813 0.2538 5.797 0.2498 5.781
0.2714 0.2676 0.2636 0.2593 0.2554
5.883 5.870 5.856 5.838 5.824
0.2662 0.2621 0.2582 0.2541 0.2499
5.845 5.830 5.815 5.799 5.782
0.2717 0.2677 0.2638 0.2597 0.2557
5.885 5.871 5.857 5.841 5.826
5.839
5.843
5.789
5.791
25 30 35 40 45
0.5025 0.4965 0.4900 0.4836 0.4768
7.763 7.758 7.752 7.745 7.738
0.5110 0.5050 0.4985 0.4925 0.4858
7.391 7.380 7.367 7.356 7.341
0.3121 0.3075 0.3028 0.2980 0.2935
6.271 6.255 6.239 6.221 6.206
0.3184 0.3139 0.3095 0.3045 0.3000
6.313 6.299 6.285 6.266 6.251
4. C8F18 0.2703 5.978 0.2662 5.962 0.2620 5.945 0.2580 5.930 0.2539 5.913
0.2759 0.2718 0.2678 0.2635 0.2596
6.019 6.003 5.989 5.971 5.957
0.2705 0.2666 0.2623 0.2583 0.2542
5.979 5.965 5.947 5.932 5.915
0.2761 0.2720 0.2679 0.2638 0.2599
6.020 6.005 5.989 5.974 5.959
5.971
5.973
5.922
5.927
25 30 35 40 45
0.5393 0.5342 0.5293 0.5245 0.5194
7.424 7.416 7.408 7.400 7.391
0.5475 0.5428 0.5375 0.5325 0.5275
7.462 7.455 7.446 7.438 7.414
0.3398 0.3358 0.3321 0.3283 0.3245
6.365 6.352 6.342 6.330 6.319
0.3460 0.3422 0.3385 0.3345 0.3305
6.403 6.393 6.382 6.370 6.349
5. C10F18 0.2943 6.067 0.2910 6.056 0.2876 6.045 0.2842 6.033 0.2810 6.023
0.2998 0.2965 0.2931 0.2900 0.2865
6.105 6.094 6.083 6.074 6.062
0.2945 0.2913 0.2879 0.2845 0.2811
6.068 6.058 6.047 6.035 6.024
0.3002 0.2968 0.2935 0.2902 0.2866
6.107 6.096 6.086 6.075 6.061
6.065
6.064
6.031
6.030
25 30 35 40 45
0.5386 0.5338 0.5293 0.5240 0.5194
7.632 7.625 7.618 7.612 7.604
0.5465 0.5420 0.5372 0.5324 0.5274
7.701 7.695 7.688 7.681 7.673
0.3392 0.3355 0.3318 0.3283 0.3245
6.542 6.532 6.521 6.512 6.501
0.3452 0.3418 0.3380 0.3345 0.3310
6.607 6.599 6.588 6.579 6.569
6. C8F16Cl2 0.2939 6.237 0.2906 6.226 0.2874 6.216 0.2842 6.206 0.2811 6.197
0.2995 0.2960 0.2929 0.2899 0.2865
6.302 6.290 6.281 6.272 6.261
0.2942 0.2909 0.2877 0.2845 0.2811
6.239 6.228 6.219 6.209 6.200
0.2996 0.2965 0.2932 0.2900 0.2868
6.302 6.294 6.283 6.273 6.263
6.260
6.258
6.229
6.228
25 30 35 40 45
0.5385 0.5337 0.5290 0.5244 0.5195
7.632 7.625 7.618 7.612 7.604
0.5467 0.5421 0.5374 0.5324 0.5275
7.670 7.665 7.658 7.651 7.644
0.3392 0.3355 0.3318 0.3283 0.3245
6.542 6.532 6.521 6.512 6.501
0.3454 0.3419 0.3381 0.3346 0.3310
6.582 6.573 6.562 6.553 6.544
7. C8F17Br 0.2939 6.237 0.2906 6.226 0.2874 6.216 0.2842 6.206 0.2811 6.197
0.2995 0.2962 0.2930 0.2899 0.2867
6.276 6.266 6.257 6.248 3.8744
0.2942 0.2909 0.2877 0.2845 0.2811
6.239 6.228 6.219 6.204 6.200
0.2995 0.2965 0.2933 0.2903 0.2869
6.276 6.232 6.268 6.259 6.250 6.200 3.8809
6.233 6.197
6422 J. Phys. Chem. B, Vol. 112, No. 20, 2008
U ) Xanl
RT γM
Ravi et al.
(7)
where R is gas constant, γ is the ratio of specific heat, M is the molecular weight, and U is the ultrasonic velocity. By feeding the experimental value of ultrasonic velocity as input, the best-fit value for y is chosen. From y the diameter is estimated using
d)
( ) 6y πFn
1/3
2.2. Estimation of Diameter Using Surface Tension and Viscosity. The peculiar property of LPFC-np is their very low surface tension σ and viscosity η when compared to any other liquids. The experimental values of η and σ were related to the radius r (2r diameter) as
[
r ) 0.5
[ ][
(8)
To extend the calculation for different temperatures, we use U ) U0 + U(T) and F ) F0 + F(T). The experimental values of U0, F, F0, T, M, and U for all seven systems were taken from ref 25. Different y and hence d values were obtained from different equations of state, namely, eqs 4, 7, and 8. Calculations were made for all seven systems mentioned in section 1.
1 Vσ1/4 7.21 × 1019 Tc1/4
r ) 0.5
√2 N
1/3
V1/3 -
]
2/5
Vf1/3 2
(9)
]
(10)
where Tc is the critical temperature and Vf ) [MU/kη]3/2. The data for critical temperature, surface tension, and viscosity for all the seven systems were available only for 25 °C and 40 °C. They were taken from the 3M corporation Web site.26 The constant k ) 4.28 × 109. By substituting the values for σ, η, and Tc the radius and hence the diameter d is estimated for all seven systems. It is compared with the diameter estimated using ultrasonic velocity.
TABLE 2: Chemical Potential (µ/kT) vs y PY temp (°C)
γ ) 1.1
SPT γ ) 1.2
MCSL
ML
γ ) 1.1
γ ) 1.2
γ ) 1.1
γ ) 1.2
γ ) 1.1
γ ) 1.2
2.4149 2.2971 2.1811 2.0542 1.9294
2.5748 2.4413 2.3257 2.1939 2.0668
2.4228 2.3049 2.1785 2.0618 1.9319
2.5748 2.4493 2.3257 2.2016 2.0744
25 30 35 40 45
12.0805 11.5164 10.9286 10.3706 9.7759
12.8151 12.2078 11.5920 11.0365 10.4388
3.4616 3.3140 3.1432 2.9791 2.8298
1. C6F14 3.6620 3.4981 3.3349 3.1724 3.0019
25 30 35 40 45
14.5342 13.8407 13.2171 12.6231 12.0331
15.3979 14.7377 14.0709 13.4363 12.7899
4.0965 3.9326 3.7627 3.6152 3.4464
2. C7F16 4.3119 4.1398 3.9814 3.8199 3.6463
2.8858 2.7715 2.6425 2.5292 2.4017
3.0449 2.9252 2.8075 2.6752 2.5345
2.8914 2.7715 2.6480 2.5319 2.4122
3.0535 2.9281 2.8075 2.6834 2.5641
25 30 35 40 45
16.4720 15.8349 15.1441 14.5438 13.8864
17.4644 16.7731 16.0799 15.3877 14.7768
4.5583 4.4046 4.2473 4.0965 3.9358
3. C8F16O 4.7948 4.6256 4.4706 4.3119 4.1398
3.2311 3.1142 3.0019 2.8858 2.7742
3.3980 3.2843 3.1665 3.0420 2.9309
3.2429 3.1229 3.0105 2.8942 2.7770
3.4070 3.2873 3.1724 3.0535 2.9394
25 30 35 40 45
17.2421 16.5606 15.8560 15.1945 14.5438
18.2624 17.5353 16.7844 16.1230 15.4184
4.7404 4.5759 4.4115 4.2473 4.0965
4. C8F18 4.9716 4.8057 4.6470 4.4706 4.3153
3.3649 3.2429 3.1200 3.0048 2.8886
3.5349 3.4100 3.2902 3.1636 3.0506
3.3709 3.2547 3.1287 3.0134 2.8973
3.5411 3.4161 3.2932 3.1724 3.0593
25 30 35 40 45
22.1955 21.4188 20.7026 20.0278 19.3387
23.5151 22.7477 21.9176 21.1671 20.4465
5.8127 5.6485 5.4996 5.3495 5.2023
5. C10F18 6.0741 5.9129 5.7590 5.5959 5.4360
4.1231 4.0141 3.9034 3.7944 3.6933
4.3085 4.1967 4.0833 3.9814 3.8800
4.1298 4.0239 3.9131 3.8040 3.6964
4.3221 4.2068 4.0965 3.9879 3.8712
25 30 35 40 45
22.0869 21.3593 20.7026 19.9590 19.3387
23.3493 22.6200 21.8717 21.1524 20.4324
5.7796 5.6323 5.4996 5.3495 5.2023
6. C8F16Cl2 6.0399 5.8961 5.7384 5.5959 5.4559
4.1065 3.9977 3.9034 3.7944 3.6933
4.2982 4.1799 4.0767 3.9781 3.8680
4.1198 4.0141 3.9131 3.8040 3.7027
4.3016 4.1967 4.0866 3.9814 3.8776
25 30 35 40 45
22.0715 21.3444 20.6596 20.0141 19.3387
23.3823 22.6360 21.9023 21.1524 20.4465
5.7879 5.6363 5.4876 5.3495 5.2023
7. C8F17Br 6.0484 5.9003 5.7425 5.5999 5.4559
4.1098 4.0010 3.8970 3.7944 3.6964
4.2982 4.1866 4.0888 3.9781 3.8744
4.1198 4.0108 3.9067 3.8040 3.7090
4.2982 4.1967 4.0899 3.9912 3.8809
Ultrasound To Characterize Fluorocarbon Interactions
J. Phys. Chem. B, Vol. 112, No. 20, 2008 6423
Figure 1. Temperature vs segment diameter.
2.3. Estimation of Chemical Potential µ/kT. The chemical potential µ/kT is related to the packing factor y and number density via the Carnahan-Starling equation of state as27
µ 8 - 9y + 3y ) ln(Fn) + y (1 - y)3 kT
2
(11)
The values of y obtained through the method outlined in section 2.1 are substituted in the above equation to estimate the chemical potential. The calculation is made for different temperatures and for different y derived from different E.O.S. 2.4. Estimation of Adiabatic βa, Isothermal βT, Compressibility, and Rao’s Constant, R. The adiabatic and isothermal compressibility of all the seven systems are calculated using the well-known relation
1 U2F γ βT ) 2 UF
βa )
(12)
The Rao’s constant is given by
1 C R) 1 l
( ∂C ∂T ) ( ∂T∂l )
P
(13)
P
We have tested the sensitiveness of the packing factor y and the diameter d to a very small change in the specific heat ratio
Figure 2. Packing fraction vs chemical potential.
γ. We have used γ ) 1.1 and 1.2. For the case of compressibility we use only 1.2. 3. Results and Discussions The entire calculation discussed from sections 2.1 to 2.4 infers that the ultrasonic velocity is the key input. The parameters estimated through the propagation of ultrasound through the system (absorption) help to characterize the interactions among the LPFC-np. To establish our argument and to make it more precise, we have obtained the packing factor y and hence the diameter d through four different equations of state and compared it with the values examined through surface tension and viscosity. They are presented in parts 1-7 of Table 1 for all seven systems. Perfluorocarbons are very similar in structure to alkanes. The fluorine atoms simply replace the hydrogen atoms. That way the segment diameter of LPFC is slightly larger than that of the normal alkanes.28 We found that the diameter estimated through the MCSL model is in good agreement with the value obtained through surface tension and viscosity invariably for all the seven systems. Hence the packing factor y obtained through this model is the best preferred. The reasons are due to the fact that the MCSL equation of state contains binomials in terms of y which reflect the weak attraction and strong repulsion.22 Hence the MCSL model is deliberately suitable for LPFC-np which naturally possess weak attraction. Therefore
6424 J. Phys. Chem. B, Vol. 112, No. 20, 2008
Ravi et al.
Figure 3. Isentropic compressibility (top) and isothermal compressibility (bottom).
the diameter and the chemical potential estimated with the help of y estimated through the MCSL model are again the best preferred. The low surface tension nature of LPFC-np accommodates remarkably the MCSL E.O.S. which accounts for weak attractions. The second reason is the unexpected low value of the packing factor y. Generally liquids possess the value between 4.5 > y > 3.5. This low value of y < 3.5 (parts 1-7 of Table 1) obtained through MCSL model reflects the gaseous nature, namely, the string affinity of LPFC-np toward gases.29 The low viscous nature of LPFC-np too backs up this interpretation.29 These arguments hold true for all seven systems. It is very interesting to note that the value of y is sensitive to γ. The diameter decreases when γ is slightly increased. They are also presented in parts 1-7 of Table 1. But this sensitivity does not influence the superiority of MCSL model over other models. The agreement is good when γ ) 1.1. The PY (virial) and SPT deviates much because the corresponding E.O.S. possess terms that depicts the strong attraction and weak repulsion.30 This is in contradiction to the nature of interaction existing in LPFC-np. Again this is true for all seven systems. The ML model is close to the MCSL model. However, we prefer to argue that MCSL is better based on the above discussions.
To have a better understanding of the variation of d with temperature T, we have plotted d versus T in Figure 1. To avoid overcrowding of graph we have presented only for the MCSL model for all seven systems. The general trend is, for a particular temperature (say 25 °C) d increases with the increase in size of the system (number of carbon atoms; parts 1-7 of Table 1), whereas d decreases with increase in temperature for every individual system Figure 1. This argument is true for all four different E.O.S. Also the diameter estimated through surface tension and viscosity corroborates nicely with this trend. The physical significance for the temperature dependency of the diameter is that at high temperature the LPFC-np have more energy and are capable of coming close to forming nano-emulsions.31 Parts 1-7 of Table 2 present the estimated value of the chemical potential µ/kT. Though we have used the CarnahanStarling E.O.S. in the expression for chemical potential, only the value of y obtained through the MCSL model is substituted. Again the value of µ/kT estimated through this model is well within the anticipated range. Here too both the PY and the SPT model deviates much. For a particular temperature (say 25 °C) the variation of µ/kT with y is linear (parts 1-7 of Table 2) with respect to increase in size of the system, whereas with the increase in temperature the value decreases, which is the expected trend32 (Figure 2).
Ultrasound To Characterize Fluorocarbon Interactions The adiabatic, isothermal compressibility and the Rao’s constant R show a linear trend with positive slope (Figure 3, top and bottom). The values are quite agreeable with the expected range. Generally, it is difficult to determine the compressibility of such nanosystems too at low volume and low concentration.33 This straightforward estimation through the ultrasonic velocity helps to overcome this difficulty. Even for a small change in the value of U there will be a considerable change in the value of β and R. To avoid over crowding of tables we have presented only the graphical representation of the variation of βa, βT, and R with respect to temperature. On the whole, the above discussions were based on the fact that the propagation of ultrasound through the entire system helps us to characterize LPFC-np interactions to a relatively good extent. 3a. Discussion on Ultrasound as a Facilitator and as a Propagator for Drug Delivery. Perfluorocarbon nanoparticles can serve as a very specific site-targeted contrast and therapeutic agent after binding to specific cellular biomarkers. Ultrasound has been used in conjunction with microbubbles to enhance the delivery of drugs and genes into cells. Ultrasonically enhanced delivery of drugs from LPFC-np to targeted cells does not involve cavitation, and hence they do not destruct the cells, namely, ultrasound has been used as a facilitator. This is mainly because of the property of LPFC-np, which allows sound at a very low speed. This kind of study is a traditional approach. In this present study, the propagation speed of ultrasound through the entire system measured at different temperatures by Marsh et al.25 is used as the key input. The thermo-physical parameters estimated above help to interpret the nature of LPFCnp that carries the drug or gene and deliver it into the targeted cell without cavitation. That way the studied systems possess a packing factor value that lies between the gas and a liquid range 2.2 < y < 3.5. And so all seven systems have low compressibility and chemical potential. This compressing nature of LPFCnp and the low value of chemical potential might allow the LPFC-np to carry the drug. Again allowing sound at low speed through the system in spite of having low packing factor is yet another reason why it does not form any cavity during the delivery of drug or gene into the cells. 4. Conclusion The extensive calculations with ultrasonic velocity as the sole input help to characterize the interactions among the LPFC-np as a drug delivering agent. The concept to use the propagation of ultrasound through the system and interpret the interactions among the LPFC-np is reasonably successful. Also a better correlation between the interaction of ultrasound with LPFC-
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