Temperature Effect on Rheological Behavior of Silicone Oils. A Model

Subscriber access provided by Kent State University Libraries

Article

Temperature Effect on Rheological Behavior of Silicone Oils. a Model for the Viscous Heating Mario R Romano, Francesca Cuomo, Nicola Massarotti, Alessandro Mauro, Mohamed Salahudeen, Ciro Costagliola, and Luigi Ambrosone J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/acs.jpcb.7b03351 • Publication Date (Web): 07 Jul 2017 Downloaded from http://pubs.acs.org on July 14, 2017

Just Accepted “Just Accepted” manuscripts have been peer-reviewed and accepted for publication. They are posted online prior to technical editing, formatting for publication and author proofing. The American Chemical Society provides “Just Accepted” as a free service to the research community to expedite the dissemination of scientific material as soon as possible after acceptance. “Just Accepted” manuscripts appear in full in PDF format accompanied by an HTML abstract. “Just Accepted” manuscripts have been fully peer reviewed, but should not be considered the official version of record. They are accessible to all readers and citable by the Digital Object Identifier (DOI®). “Just Accepted” is an optional service offered to authors. Therefore, the “Just Accepted” Web site may not include all articles that will be published in the journal. After a manuscript is technically edited and formatted, it will be removed from the “Just Accepted” Web site and published as an ASAP article. Note that technical editing may introduce minor changes to the manuscript text and/or graphics which could affect content, and all legal disclaimers and ethical guidelines that apply to the journal pertain. ACS cannot be held responsible for errors or consequences arising from the use of information contained in these “Just Accepted” manuscripts.

The Journal of Physical Chemistry B is published by the American Chemical Society. 1155 Sixteenth Street N.W., Washington, DC 20036 Published by American Chemical Society. Copyright © American Chemical Society. However, no copyright claim is made to original U.S. Government works, or works produced by employees of any Commonwealth realm Crown government in the course of their duties.

Page 1 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

The Journal of Physical Chemistry

Temperature Effect on Rheological Behavior of Silicone Oils. A Model for the Viscous Heating Mario R Romano MD PhD,† Francesca Cuomo,‡ Nicola Massarotti,¶ Alessandro Mauro,¶,§ Mohamed Salahudeen ,¶ Ciro Costagliola,k and Luigi Ambrosone∗,k †Department of Biomedical Sciences, Humanitas University, via Manzoni 113, 20089, Rozzano, Milan, Italy ‡Department of Agriculture, Environment and Food Sciences, Molise University, 86100 Campobasso, Italy ¶Department of Engineering, Universitá degli Studi di Napoli ”Parthenope”, Centro Direzionale, Isola C4, 80143 Napoli §Universitá telematica Pegaso, Piazza Trieste e Trento n.48, 80132 Napoli kMedicine and health science,”Vincenzo Tiberio”, Molise University, 86100 Campobasso, Italy E-mail: [email protected]

1 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Abstract Rheological behavior of silicone oils,(CH3 )3 SiO−[Si(CH3 )2 O]n −Si(CH3 )3 , and their mixtures is studied. Shear stress measurements, in the temperature range 293-313 K, reveal that this family of polymers are shear thinning liquids with a yield stress below which no flow occurs. Experimental diagrams, i.e. shear stress versus shear rate, are satisfactorily described by the fluid model of Casson over a wide range of shear rates. In order to monitor the effect of temperature on fluid properties, Casson’s rheological model is reformulated using the fictitious shear rate, γ˙ f , and the infiniteshear viscosity, η∞ , as constitutive parameters. Due to low intermolecular forces and high chain flexibility, γ˙ f varies very little when the temperature increases. For this reason, the apparent material viscosity depends on temperature only through η∞ , which exponentially decreases until high shear rates are reached and there is more alignment possible. Interestingly, the temperature sensitivity of this pseudoplastic behavior is the same for all the silicone oils investigated, therefore they can be classify according to their tendency to emulsify. Experimental results are then used to model the flow of silicone oils in a cylindrical pipe and estimate the temperature increase due to viscous heating. Numerical results show that the normalized temperature, i.e. ratio of fluid temperature to wall temperature, increases of approximately 23% and the apparent viscosity decreases drastically, going towards the center of the tube. The non-Newtonian nature of fluid is reflected in the presence of a critical region. In this region, the velocity and temperature gradients vanish. Since silicon oil is a surgical tool, we hope that acquired physicochemical information can provide a help to facilitate the removal of this material during surgical procedure.

Introduction Silicone oils consist of an inorganic siloxane backbone with organic pendant organic groups usually methyl. 1 This chemical structure is responsible for their low intermolecular forces, and their wide use in many applications. Of products commercially used, the polydimethyl2 ACS Paragon Plus Environment

Page 2 of 26

Page 3 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


siloxanes (PDMS) with general chemical formula (CH3 )3 SiO−[Si(CH3 )2 O]n −Si(CH3 )3 , are used in ophthalmology as surgical tools. 2 However, in-vivo studies have shown that silicone oil often becomes emulsified, i.e. toxic, if left in eye for a long period of time. For this reason, it was proposed to add a small portion of high-molecular mass PDMS molecules to conventional silicone oil to increase the resistance to emulsification, 3 or to use perfluorocarbons with density higher than water. 4 This silicon oil is easier to inject and remove when compared to conventional silicone oil with the same shear viscosity. 3 Nevertheless, the problem of silicone oil emulsification remains and moreover a prolonged use of the perfluorocarbon liquid leads to irreversible damages on the retina. 5,6 In addition, temperature fluctuations, occurring during vitreoretinal surgery, have been shown to be statistically significant and affect the same surgical procedure. 7 Recently with the advent of finer instruments and cannulae with diameter of 0.6-0.4 mm, there is even more need for less viscous silicon oil. On the other hand, a less viscous oil would be more likely to emulsify. These situations motivated us to investigate silicone oils under shear-stress. We are interested in developing a quantitative correlation to describe how the flow-properties of the material are influenced by temperature changes. In the near future, we aim to change the rheology of the polymer and its shear viscosity, keeping high strain rate extensional viscosity. The paper is divided into two parts. The first of these concerns the experimental determination of shear stress and shear rate as a function of temperature. The suitability of several rheological models are tested in order to find the actual non-Newtonian behavior of silicone oils and perfluorurate compounds. In the second part, experimental results are exploited to model the steady flow of silicone oils in a cylindrical pipe to establish the temperature profile induced by viscous heating. The proposed model is a starting point for understanding the ability of spontaneously emulsify exhibited by these compounds. 8



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Materials and Methods Chemicals All the materials used in this study were supplied by Fluoron GmbH (Ulm, Germany). These are commercial products used in vitreoretinal surgery. This is because the experimental results will provide enough information to establish surgical equipment design. Siluron 1000 and siluron 5000 are pure PDMS of chemical formula (CH3 )3 SiO[(CH3 )2 SiO]n Si(OCH3 )3 and molecular mass(MM) 3700 and 65000 Da, respectively. Siluron 2000 is a mixture of 95% Siluron 1000 and 5% of PDMS 423000 Da. Siluron Xtra is a mixtures consisting of 95% siluron 1000 and 5% PDMS of very high MM and kinematic viscosity 2.5 · 106 cSt. Densiron 68 is a blend of 60.5% siluron 5000 and 30.5% of perfluorohexyloctane (F6H8)

Rheological measurements Rheological analyses were performed in triplicate using a stress control rheometer (Haake MARS III Thermo Scientific) equipped with a cone-plate geometry and cone plate angle 1 (gap 53 µm with diameters 20 mm), operating in both steady and oscillation mode. The selected temperatures of 20, 23, 25, 27, 30, 33, 35, 37 and 40◦ C were controlled by the Peltier device with an accuracy of 0.1◦ C. For the dynamic oscillatory rheology investigation, the samples were exposed to increasing stress (0.01-100 Pa) at a constant frequency (ν = 0.1 or 1 Hz) to determine the linear viscoelastic range of the samples. In any case, none of the compounds tested exhibited viscoelastic behavior.

Computational procedure When a real fluid, originally at equilibrium, is subjected to a stress, its molecules will change shape and flow. 9 In a very short time after the application of the stress, every molecule suffers a displacement from its initial position to a new equilibrium position, and the work exerted on the fluid is stored as free energy of deformation. If the stress is removed, the fluid tends 4 ACS Paragon Plus Environment

Page 4 of 26

Page 5 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


to recover its configuration, but the recovery is incomplete as the result of the appearance of viscous flow. Of course the elucidation of the basic mechanism can be attained by studying how the viscous resistance is related to the molecular structure. This approach has received a great deal of attention so that numerous models have been suggested to relate the shear stress, τ , to shear rate, γ. ˙ Herein, the Power Law, 10 Herschel-Bulkley, Mizrahi-Berk, 11 and Sisko 12 models were applied to the rheological data in order to fit the best representation of function τ = f (γ). ˙ Obviously, a criterion for the selection must be adopted. We chose as best curve the model that exhibited the highest correlation coefficient, R, and the lowest χ2 . 13 It is noteworthy that the Mizrahi-Berk model fits the data with greater accuracy than other models (R = 0.99998, χ2 = 0.1). The equation adopted was √

τ = k1 + k2 γ˙ n

(1)

where k1 , k2 and n are adjustable parameters. 14 The non-linear numerical regression to fit data to eq 1 is performed by the Levenberg-Marquardt method which gradually shifts the search for the minimum χ2 from steepest descent to quadratic minimization (GaussNewton). 15 Interestingly, this procedure applied to all silicone fluids, gives n = 0.502±0.002. Based on this finding we rewrite eq (1) as p √ τ = k1 + k2 γ˙

(2)

Eq 2 is the well-known Casson equation 16 which relates the stress square root to the square √ √ root of the shear rate. One important feature of the Casson model is that τ vs. γ˙ is a straight line, and the parameters k1 , k2 can be calculated through a linear least square. Figure 1 shows, as an example, the square root of shear stress measured at 20◦ C for all √ silicone oils, as a function of γ. ˙ The straight lines confirm the prediction based on Casson model, therefore the parameters k1 , k2 for each temperature and for all tamponade agents were evaluated via linear least squares and are collected in the form of table as electronic 5 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 6 of 26

supplementary information.

Results and Discussion Rheological behavior Yield stress The constants, k1 , k2 , which appear in Casson model reflect the non-Newtonian behavior of the fluid. To understand the physical significance of the parameter k1 we consider the limit of τ for γ˙ →0 in eq 2 lim τ = τ0 = k12

γ→0 ˙

(3)

Thus the quantity k1 is a measure of yield stress, namely it is the finite amount of stress required to initiate flow. It is a critical level of stress which has to be attained in order to initiate the flow. From the practical standpoint, this stress can be considered to impart stability to the materials in low-shear conditions, such as storage. Therefore, the material begins to flow only when the deformation induced by the shear exceeds the force between molecules. In applying these concepts to the data reported as electronic supplementary information, it is clear that the silicone oils, with the exception of the Siluron Xtra, exhibit values of τ0 in the range of 0.1 − 0.2 Pa, indicating that the intermolecular attraction forces have the same effect. The situation experienced by Siluron Xtra molecules is quite different (τ0 = 0.50-0.56 Pa), this is because the presence of long lifetime junctions produces a straining of the chains approaching of a permanent network. Experimental results suggest that k2 > 0 so that eq 2 becomes √

τ−

p √ τ0 = k2 γ˙ > 0


(4)

Page 7 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Such a relationship has physical significance only for τ ≥ τ0 . It follows that for silicone oils the constitutive equation is described by  √ √ ( τ − τ0 )2    k22 γ˙ =    0

for

τ ≥ τ0 (5)

for

0 ≤ τ < τ0

It is evident from eq 5 that the parameter τ0 represents a lower stress strain of the fluid and it identifies a critical region where the shear rate remains constant.

Apparent viscosity In many non-Newtonian systems the viscosity is a function of the rate of deformation, therefore the apparent viscosity is defined as the ratio τ /γ. ˙ By applying this definition to eq 2, one finds 2 τ k1 2 η = = k2 1 + √ γ˙ k2 γ˙

(6)

It is apparent that a finite γ˙ value triggers a shear-thinning behavior, namely the viscosity decreases with increase in shear-rate. With increase in shear-rate, more polymer chains are being disentangled and oriented, opposing a smaller resistance to the flow, thus lowering viscosity. The limit of eq 6 for γ˙ → ∞, yields

lim η = η∞ = k22

γ→∞ ˙

(7)

This means that k22 is the viscosity in the configuration of maximum alignment between the polymer chains and flow. On the other hand, from dimensional analysis of eq 6 one realizes that the ratio k1 /k2 represents a shear rate, ie p k1 p = τ0 /η∞ = γ˙f k2


(8)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 8 of 26

It is a fictitious shear rate of a fluid with viscosity η∞ subjected to shear stress τ0 . p In Fig 2, γ˙ f is plotted as a function of T in the range 293−313 K, for all silicone oils. These exhibit an increasing linear trend, except siluron 2000, whose trend is very slightly decreasing. p Experimental data were fitted by the relationship γ˙ f = a + bT and the parameters a, b determined via a linear regression and collected in Tab 1. It is noteworthy to emphasize that p a < 1 and b 1, so that γ˙ f assumes practical significance only for extremely small shear rate. In order to explain such a behavior we must take into account that γ˙f depends on τ0 and η∞ . Yield stress, i.e. τ0 , is indicative of degree of shear thinning and it is expected to decreases as the temperature rises. As far as η∞ is concerned, it is obvious that no material actually experiences an infinite shear, but it is a good representation of the condition where all rheological structures have been broken down. Due to the pseudoplastic nature of the material, η∞ decreases with the temperature. Therefore, the very small slope of the straight lines displayed in fig 2 indicates that τ0 and η∞ decrease with comparable rates and their contributes compensate each other in the ratio. On the other hand, when a silicone oil, originally at equilibrium, is subjected to a stress the equilibrium position of polymer is deformed and every molecules suffers a displacement from initial position. This happens in the relaxation time, i.e. a very short time after the application of stress. This explains the p numerically small values of γ˙ f . Both effects have to be linked to weak intermolecular forces p and high chain flexibility. Therefore, at higher shear rate, the ratio γ˙ f /γ˙ is negligible with respect to the unit and the apparent viscosity is well-described by η∞ (T ). In fig 3, a semilog plot of η∞ versus T displays a linear trend with negative slope, i.e.

lg η∞ = c − dT

or

η∞ = Ae−dT

(9)

where c, A and d are constants. The constant d indicates the temperature sensitivity of the polymers. It is immediately seen from fig 3 that all the lines are parallel, accordingly d is the same for all investigated samples. When γ˙ → ∞ all molecules have reached the preferred


Page 9 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


orientation occurring during flow, so that their flow capacity is virtually the same for all polymers. Thus it is possible to write the viscosity function as

η(γ, ˙ T ) = Ae

−dT

a + bT 1+ √ γ˙

(10)

With the aid of the parameters of Table 1, this equation can be used to predict the rheological behavior of the silicone oils in the temperature range 20-40◦ C. Thus, on the basis of the experimental results, this function is satisfactorily simplified into

η(γ, ˙ T ) = Ae−dT

(11)

Such experimental evidence implies that the temperature effect on the apparent viscosity of a silicone oil can be assessed knowing d and a reference viscosity, ie

log

η = −d(T − TR ) ηR

(12)

The main property of eq 10 is that it provides the opportunity to estimate the tendency of an oil to emulsify and disperse into droplets over time. 17 Indeed, the less viscous is the substance, the lower the energy that is required to disperse a large bubble of silicone oil into small droplets. Through eq 10, it is established that the tendency to emulsify follows the following order Siluron 5000< Siluron Xtra < Siluron 2000< Densiron 68< Siluron 1000

Temperature and velocity profiles In the previous section we discussed the non-Newtonian rheological behaviour of silicone oils. We, now, continue the discussion by applying the experimental results in order to study the flow of such oils through a cylindrical pipes and to estimate the raising temperature of fluids due to viscous heating. 9 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 10 of 26

Let us consider an axially symmetric, laminar, steady and fully developed flow of nonNewtonian incompressible viscous fluid in the axial direction x. We assume a system of cylindrical polar coordinates as displayed in Fig 4. From these hypotheses one may derive that conservation of total mass, conservation of energy and conservation of moment equations, reduce to the following set of differential equations ∂p 1 d + rτrx = 0 − ∂x r dr ηλ d dT 2 r + τrx =0 r dr dr

(13) (14)

where λ is the thermal conductivity, assumed to be constant, p is the pressure, τrx is the components of the stress tensor, ux is the x-component of velocity. A first integration of eq 13 with the condition that τrx is finite at r = 0, yields

τrx

∂p r = − ∂x 2

(15)

Accordingly, the wall stress becomes τRx =

∂p R − ∂x 2

(16)

The simplest way to reduce eqs 13-14 to a more useful form is to write them in a dimensionless manner, through appropriate scale factors. For a good choice of the scale factors, we notice that for liquid polymers of high viscosity and low thermal conductivity a non negligible temperature gradient is generated. This is because the heat generated by viscous heating is not removed at sufficient rate and the temperature of the liquid flowing near the center of the pipe may be appreciably greater than the wall temperature, TR . It is only natural, then, to choose τRx , R, and TR as references to re-scale the shear stress, the radial distance and


Page 11 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


temperature, respectively ie

τˆ =

τrx ; τRx

r ; R

ξ=

T Tˆ = TR

(17)

Now, we rewrite eq 13 in dimensionless form

τˆ =

φ ξ 2

(18)

where the dimensionless number φ is defined by ∂p R φ= − ∂x τRx

(19)

The substitution of eq 18 into eq 14, where all variables have been reduced, yields ˆ d dT Br 3 ηˆ ξ =0 ξ + φ2 dξ dξ 4

(20)

where Br is the Brinkman number 18 2 4R2 τRx λTR ηR

Br =

(21)

Making use of eq 16 one finds 2 ∂p R − ∂x 4

φ = 2 ; Br =

4λTR ηR

(22)

It is not difficult to show that these relationships are equivalent to choose as twice the maximum of the corresponding isothermal Poiseuille flow, i.e. R Uc =

2

∂p − ∂x

2ηR 11

ACS Paragon Plus Environment

(23)


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 12 of 26

Such a choice requires that the pressure is used as independent parameter, namely for a given pipe and silicone oil, R and ηR are given. In order to calculate the temperature profile in the fluid through eq 20, the function ηˆ(ξ) has to be made explicit. Of course, the particular shape adopted for this function depends on the physicochemical properties of the fluid. On the other hand, if one reminds that to a body deformation corresponds a material displacement, it is clear that a shear rate is equivalent to a velocity gradient, in briefly γ˙ = −dux /dr. Accordingly, eq 9, experimentally ascertained, is the constitutive equation of the material, furthermore the viscosity function is satisfactorily described by eq 12 which allows is to be considered the dimensionless of viscosity. Therefore, eq 20 becomes ξ

d2 Tˆ dTˆ Br 3 χTˆ + + χξ e =0 dξ 2 dξ e

(24)

where χ = dTR is a dimensionless number. A discussion on the properties of the solutions of this equation, including their stability 19 will be object of study for a subsequent paper, herein numerically solve the eq 24 with boundary conditions Tˆ(1) = 1

dTˆ = 0 at ξ = ξ0 dξ

and

(25)

The first condition means that the wall temperature is kept constant, the second condition is a consequence of the Casson model, where velocity and temperature gradients are zero in the critical region. In this regard, we note that eqs 15-16 state that the ratio of the stress at distance r to the stress at the wall is equal to the ratio of the radial position and pipe radius. This implies that the critical region, 0 ≤ τˆ ≤ τˆ0 due to the yield stress, can be converted into an equivalent physical region in the pipe near the center of the cylinder, 0 ≤ ξ ≤ ξ0 . Once determined the function Tˆ(ξ), it is used to calculate velocity by means of eq 18

−χ

Z

uˆx (ξ) = e

1

ˆ

ξeχT (ξ) dξ

for 0 ≤ ξ ≤ ξ0

ξ


(26)

Page 13 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


ˆ assumptions must be made about the parameters To compute the numerical values of T (ξ), χ, ξ0 and Br. Experimentally it was found that the parameter d is common to all of silicone oils investigated, therefore fixed TR = 293, χ = 5.86 is uniquely determined, in addition we assume ξ0 = 0.1. Having obtained numerical values for the constants χ and ξ0 it is possible to derive numerically the function Tˆ(ξ) with varying the parameter Br. The Richardson extrapolation technique 20 was adopted to solve eq 24 and results are illustrated in Fig 5. A thorough analysis reveals some interesting aspects. Firstly, near the center of the tube there is a discontinuity induced by non-Newtonian nature of fluid. Secondly, a critical Br exists, beyond which the system does not allow any solution of steady state. The limit value is found to be Brc = 1.3652, it is a manifestation of non-linearity of eq 24. These two features make the silicone oils especially prone to the formation of convective motions, or also to the spontaneous generation of emulsions. 8 To understand this aspect we refer to Fig 6 which shows velocity profiles and of the viscosity function for different values of the parameter Br. As it can be seen, going from the wall to the center of the pipe velocity increases while the viscosity decreases. For low velocity, dˆ ux /dξ may increase more rapidly than the effect of viscosity decrease and the stress grows linearly (eq 18). However, it can locally occur that the velocity increase is so high that it cannot be compensated by the decrease of viscosity, therefore the shear stress (that is the product of viscosity and velocity gradient) decreases. On the other hand, if the shear stress becomes comparable to the yield stress the fluid, because of its low surface tension, reacts dispersed into small droplets.

Conclusions From rheological measurements performed on silicone oils one may conclude that regardless of whether they are pure or a mixture, do not exhibit any viscoelastic behaviour and their non-Newtonian behavior is fully described by the Casson fluid model. The intrinsic param-



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

eters of the model are based on two physico-chemical properties of the fluid. The first is the yield stress, linked to the finite amount of stress needed to initiate flow, the second one is the limit viscosity due to the alignment of the polymer chains to the shear. To evaluate the dependence of viscosity on temperature is introduced a fictitious shear rate, which varies very slowly with T . On the contrary, η∞ exhibits an exponential decrease with temperature. With good accuracy the temperature dependence of the apparent viscosity may be attributed to its limit value. Interestingly, the experiments show that the average decrease, as measured by the parameter d, is equal for all silicone oils investigated. Temperature gradients may be caused by internal heat sources, although the boundaries are isothermal. The effect of viscous heating was investigated by simulating the flow of the silicone oil through a cylindrical tube with fixed wall temperature. Results indicate that a stationary solution of the motion exists and the temperature, near the center of the tube, increases 5-23% depending on the Br value. Non-Newtonian nature of the fluid is manifested in the presence of a critical region where velocity and temperature gradients vanish. In the vicinity of the critical region, the velocity exceeds the maximum of the Poiseuille flow, with drastic reduction of the apparent viscosity. The non-linearity of the motion equations implies the existence of a threshold number, BRC which identifies the region over which the system does not admit steady solutions. In this region the system may become unstable and take an unpredictable configuration. As we have already mentioned several times, silicone oils are widely applied clinically as intraocular tamponades, however they emulsify if left in the eye for a long time period, becoming toxic for ocular tissue. For optimum emulsion stability in a given system the rate that equilibrium surface tension is attained by a freshly formed surface should be within certain limits. 21 Therefore, if silicone oil confers film elasticity, it acts as an emulsifier; on the contrary if silicone oil reduce film elasticity, it acts as a demulsifier. Which of the two ways takes the tamponade depends on several factors including the viscosity, temperature and mechanical energy supplied to the fluid from surgical instruments. With this study,


Page 14 of 26

Page 15 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


an attempt has been made to examine the relationship between molecular structure and rheological-thermal behavior, for silicones, particularly the polydimethylsiloxanes and their mixtures. What does the future hold for this research? One can safely predict a broadening of properties described in this paper to so-called heavy tamponades, i.e. semifluorinated alkanes. One development that can be expected is a correlation between temperature, viscosity and emulsifying tendency. Since silicone oil is a surgical tool that need to be introduced and removed through very fine instruments an cannulae, we hope that physical chemistry studies can get us enough information because, in the near future, we can change the rheology of the polymer and its shear viscosity, keeping high strain rate extensional viscosity.

Acknowledgement The authors gratefully acknowledge the financial support of TeVR SIR project n. RBSI149484, CUP E62I15000760008 and Fluoron GmbH for supplying of tamponades utilized in this study.

Supporting Information Available Tables of Casson parameters and fictitious velocities, obtained by fitting experimental data on silicone oils at different temperatures in the range (20-40◦ C)

References (1) Li, Q.; Han, X.; Hou, J.; Yin, J.; Jiang, S.; Lu, C. Patterning Poly(dimethylsiloxane) Microspheres Via Combination of Oxygen Plasma Exposure and Solvent Treatment. J Phys Chem B 2015, 119, 13450–13461. (2) Williams, R.; Kearns, V.; AC, L.; Day, M.; Garvey, M.; Krisna, Y.; D, M.; Strappler, T.; 15 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Wong, D. Novel Heavy Tamponade for Vitreoretinal Surgery. Invest Ophthalmol Vis Sci 2013, 54, 7284–7492. (3) Zhao, Z.; An, S.; Xie, H.; Han, X.; Wang, F.; Jiang, Y. The Relationship Between the Hydrophilicity and Surface Chemical Composition Microphase Separation Structure of Multicomponent Silicone Hydrogels. J Phys Chem B 2015, 119, 9780–9786. (4) Chan, Y.; Williams, R.; Wong, D. Flow behavior of Heavy Silicone oil During Eye Movements. Invest Ophthalmol Vis Sci 2014, 55, 8453–8457. (5) Tamai, K.; Toumoto, E.; Majima, A. Protective Effects of Local Hypothermia in Vitrectomy under Fluctuating Intraocular Pressure. Exp Eye Res 1997, 65, 733–738. (6) Frink, M.; Floché, S.; van Griensven, M.; Mommsen, P.; Hildebrand, F. The Impact of Hypothermia on Molecular Mechanisms Following Major Challenge. Mediators Inflamm 2012, Article ID 762840, 1–13. (7) Romano, M. R.; Romano, V.; Mauro, A.; Angi, M.; Costagliola, C.; Ambrosone, L. The Effect of Temperature Changes in Vitreoretinal Surgery. Translational Vision Science and Technology 2016, 5, 261–273. (8) Costagliola, C.; Semeraro, F.; Dell’Omo, D.; Zeppa, L.; Bufalo, G.; M Cardone, a. M. R. R.; Ambrosone, L. Some Physicochemical Remarks on Spontaneous Emulsification of Vitreal Tamponades. BioMed Research International 2014, Article ID 243056, 1–6. (9) Ambrosone, L.; D’Errico, G.; Sartorio, R.; Vitagliano, V. Analysis of Velocity CrossCorrelatuon and Preferential Solvation for the System N-Methylpyrrolidone-Water at 20◦ C. J Chem Soc Farad Trans 1995, 91, 1339–1344. (10) Nayef Mo Ghasem,; Al-Marzouqi, M. H.; El-Naas, M. H. Effect of Temperature, Composition, and Shear Rate on Polyvinylidene Fluoride/Dimethylacetamide Solution Viscosity. J Chem Eng Data 2009, 54, 3276–3280. 16 ACS Paragon Plus Environment

Page 16 of 26

Page 17 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


(11) Cvek, M.; Mrlik, M.; Pavlinek, V. A Rheological Evaluation of Steady Shear Magnetorheological Flow Behavior Using Three-Parameter Viscoplastic Models. J Rheol 2016, 60, 687–694. (12) Yen Na, T.; Hansen, A. G. Radial Flow of Viscous Non-Newtonian Fluids Between Disks. Int. J. Non-Linear Meckcmics 1967, 60, 261–273. (13) Bufalo, G.; Ambrosone, L. Method for Determining the Activation Energy Distribution Function of Complex Reactions by Sieving and Thermogravimetric Measurements. J Phys Chem B 2016, 120, 244–249. (14) Jr Dennis, J. E.; More, J. J. Quasi-Newton Methods, Motivation and Theory. SIAM Rev 1977, 19, 46–89. (15) Di Nezza, F.; Guerra, G.; Coastagliola, C.; Zeppa, L.; Ambrosone, L. Thermodynamic Properties and Photodegradation Kinetics of Indocyanine Green in Aqueous Solution. Dyes and Pigments 2016, 134, 342–347. (16) Mernone, A.; Mazumdar, J.; Lucas, S. A mathematical Study of Peristaltic Transport of a Casson Fluid. Mathematical and Computer Modelling 2002, 35, 895–912. (17) Ambrosone, L.; Ceglie, A.; Colafemmina, G.; Palazzo, G. Emulsions: A Time-Saving Evaluation of the Droplets’ Polydispersity and the Dispersed Phase Self-Diffusion Coefficient. Langmuir 1999, 15, 6775–6780. (18) Coelho, P. M.; Pinho, F. T. A Generalized Brinkman Number for Non-Newtonian Duct Flows. J Non-Newtonian Fluid Mechanics 2009, 156, 202–206. (19) Ambrosone, L. Double-diffusive Instability in Free Diffusing Layer: a General Formulation. Physica B 2000, 292, 136–152. (20) Zlatev, Z.; Fraagó, I.; Havasi, A. Richardson Extrapolation Combined with the Sequential Splitting Procedure and θ-Method. Cent Eur J Math 2012, 10, 159–172. 17 ACS Paragon Plus Environment


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

(21) Mosca, M.; Dianton, A.; Lopez, F.; Ambrosone, L.; Ceglie, A. Impact of Antioxidants Dipersions on the Stability and Oxidation of Water-in-Olive Oil. Eur Food Res &Tech 2013, 236, 319–328.


Page 18 of 26

Page 19 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Table 1: Parameters for calculating the viscosity of the silicone oils as a function of both temperature and shear rate. Compound Siluron 1000 Siluron 5000 Siluron 2000 Siluron Xtra Densiron 68

a b A d s0.5 s0.5 K−1 Pa·s K−1 -0.68±0.06 0.0027±0.0005 0.57±0.07 0.022±0.002 -0.23±0.05 -0.0010±0.0002 0.91±0.05 0.018±0.002 0.70±0.01 0.0017±0.0005 0.11±0.01 0.014±0.02 -0.77±0.07 0.0035±0.0008 1.6±0.1 0.021±0.02 -0.14±0.09 0.0010±0.0004 0.68±0.04 0.021±0.02



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 1: Test to validate the Casson model for (a) Siluron 1000, (b) Siluron 2000, (c) Siluron 5000, (d) Siluron Xtra, (e) Densiron 68, at 20◦ C.


Page 20 of 26

Page 21 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 2: Root square of the fictitious rate as a function of temperature, for Siluron 1000 (), Siluron 2000(4), Siluron 5000(◦), Siluron Xtra(•), Densiron 68 ( ).



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 3: Limit shear stress (γ˙ → ∞) as a function of temperature for Siluron 1000 (), Siluron 2000 (4), Siluron 5000 (◦), Siluron Xtra (•), Densiron 68 ( ).


Page 22 of 26

Page 23 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 4: Limit shear stress as a function of temperature



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Figure 5: Temperature distribution in a pipe due to to viscous heating and non Newtonian rheological behavior of silicone oils. Simulation has been performed using α = 0.1, d = 0.02 K−1 and wall temperature TR = 293 K. Numbers on curves are values of Brinkman number. The red curve identifies the limit of the steady flow.


Page 24 of 26

Page 25 of 26

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


Figure 6: Illustration of the flow of silicone oils, satisfying the Casson model, through a cylindrical duct in terms of dimensionless property. The fluid is subject to a viscous heating due to the viscosity variation with temperature. It is interesting to note how opposite directions of the Brinkman number indicate increasing profiles of velocity and viscosity.



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Table of Content Graphic

Figure 7


Page 26 of 26

Temperature Effect on Rheological Behavior of Silicone Oils. A Model

Recommend Documents