Dextran Aqueous Mixtures

Journal of Colloid and Interface Science 253, 367–376 (2002) doi:10.1006/jcis.2002.8572

Interfacial Tension in Phase-Separated Gelatin/Dextran Aqueous Mixtures P. Ding,∗ B. Wolf,† W. J. Frith,† A. H. Clark,† I. T. Norton,† and A. W. Pacek∗,1 ∗ School of Chemical Engineering, The University of Birmingham, Birmingham, Edgbaston B15 2TT, United Kingdom; and †Unilever Research, Colworth Laboratory, Sharnbrook, Bedford MK44 1LQ, United Kingdom Received January 15, 2002; accepted June 28, 2002; published online August 27, 2002

Interfacial tensions in aqueous polymer/polymer phaseseparated mixtures used in bioseparation (1) have been found to be in the range σ ⊂ 0.5–500 µN/m, i.e., two to three orders of magnitude lower than in typical oil/aqueous systems (2). The effect of the concentration of biopolymers on interfacial tension in polyethylene glycol (PEG)/dextran mixtures (frequently used in bioseparation) has been investigated by Bamberger et al. (3) and Forciniti et al. (4). They reported interfacial tensions in the range σ ⊂ 30–300 µN/m depending on the molecular weight and total concentration of both polymers and found that experimental data can be correlated either with the length of the tie-line (TLL),

The effect of solute concentrations on interfacial tension was investigated in phase-separated mixtures of dextran and gelatin over a range of concentrations that covered different tie-lines and different positions on one tie-line. The investigations were carried out using equilibrated gelatin-rich and dextran-rich phases in a computer-controlled Couette device at 40◦ C (above the gelation point of gelatin) and interfacial tensions were measured using the retracting drop method. The results show that the interfacial tension can be related to the length of the tie-line or to the difference in the concentration of dextran (or gelatin) in the separated phases. Interfacial tension increases as either of these parameters increases. For concentrations lying on any single tie-line, the interfacial tension is constant and independent of the concentration of biopolymers. Also, the addition of small amounts of low molecular weight dextran to a dextran-rich phase does not significantly affect the interfacial tension between the gelatine-rich and dextran-rich phases. Experimental results were also compared with theoretical predictions of the interfacial tension using a Flory–Huggins based analysis of the measured tie-line data. Reasonable agreement was found between predicted and measured values, indicating that this approach captures the basic physics of the system. C 2002 Elsevier Science (USA) Key Words: interfacial tension; aqueous/aqueous two-phase systems; tie-lines; retracting drop method.

σ = C1 · (TLL)C2

[1]

σ = C3 · exp[C4 · (TLL)],

[1a]

or with the difference between the concentrations of the polymers in the separated phases: σ = C5 · (X dex )C6

[2]

σ = C7 · (X PEG ) .

[2a]

C8

In the above equations C1 to C8 are experimental constants that depend on the type of biopolymer, molecular weight, temperature, and concentrations. Forciniti et al. (4) analyzed the accuracy of the above equations and dismissed the applicability of Eq. [1a] as it gives a finite value of interfacial tension when the length of the tie-line is equal to zero (critical point), and found that all other equations gave the same accuracy. There are very few data on interfacial tension in the aqueous/ aqueous phase-separated systems used in the food industry. Guido et al. (5) measured the interfacial tension in the Naalginate/Na-caseinate system with a viscosity ratio λ = 10 (Naalginate-rich phase dispersed, µd = 2.5 Pa, µc = 0.25 Pa) and a viscosity ratio λ = 0.65 (Na-caseinate-rich phase dispersed, viscosity of phases not given) and in both cases obtained practically the same values of σ = 8.15–8.41 µN/m. For a similar system, but with low molecular weight Na-alginate, and a viscosity ratio λ = 10 (Na-alginate-rich phase dispersed, µd = 0.25 Pa, µc = 0.025 Pa) Pacek et al. (6) reported σ = 1.1 µN/m. Recently

1. INTRODUCTION

Phase-separated protein/polysaccharide mixtures are frequently encountered in a broad range of food products from ice cream to low-fat spreads. It is consequently very important to understand the phase behavior and factors affecting phase morphology in such systems if we are to improve our ability to manufacture and design foods. The quality of such food products often depends on the morphology and structure of these mixtures, which in turn strongly depends on the interfacial tension between the two phases. However, until recently, there have been very few studies of the interfacial tension in the aqueous/aqueous phase-separated systems used in the food industry.

1 To whom correspondence should be addressed. Fax: ++44 121 414 5324. E-mail: [email protected].

367

0021-9797/02 $35.00

 C 2002 Elsevier Science (USA)

All rights reserved.

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DING ET AL.

Wolf et al. (7) measured the interfacial tension of the gellan/κcarrageen system (λ = 2.8, gelan-rich phase dispersed, µd = 1.6 Pa, µc = 0.57 Pa) and reported σ = 7.5 µN/m. In this paper a systematic investigation of the effect of the concentration of gelatin and dextran on the interfacial tension in phase-separated gelatin/dextran systems is reported. This system has been selected because it is a good model of protein/polysaccharide mixtures commonly found in food systems and a number of aspects of this mixture have been studied previously (8). The interfacial tension was measured using a modified retracting drop method (6) at three different compositions corresponding to three different tie-lines and also at two compositions on the same tie-line. The applicability and accuracy of this method for the measurement of extremely low interfacial tensions is discussed in the context of traditional measuring methods.

torted drops/threads to be obtained at relatively high frequency, allowing the interfacial tension to be calculated (6) with sufficiently high accuracy. For small deformations (D  1), the deformation parameter changes exponentially with time,  t D(t) = D0 exp − , τ

[3]

and for a single drop (or in dispersions of low volume fraction of the dispersed phase) the characteristic time (τ ) depends on the interfacial tension, the viscosity of both phases, and the drop radius (5, 17): τ=

µc R0 (19λ + 16)(2λ + 3) . σ 40(λ + 1)

[4]

Equation [3] can be rearranged to: 2. TECHNIQUES USED FOR THE MEASUREMENT OF INTERFACIAL TENSION

The techniques used to measure interfacial tension can be broadly classified into three groups (9, 10): (a) techniques in which the interfacial tension is measured/ calculated from the balance between gravitational forces and surface forces, (b) techniques based on the measurement of the equilibrium shape and size of drops, (c) dynamic methods based on direct observation of the interface between the phases which involves the analysis of (i) the deformation and breakage of a thin thread of one phase (10, 11), (ii) the steady-state shape of drops in the system subjected to a well-controlled flow field (12), (iii) the retraction of a deformed drop after cessation of flow (10, 13). The selection of the technique depends on the physical properties of both liquids, mainly their density and viscosity. In aqueous/aqueous two-phase systems the density of both phases is very similar and the viscosity of at least one phase (and very often both phases) is rather high. As the interfacial tension in these systems is extremely low, application of techniques belonging to groups (a) and (b) either is impossible (6) or can lead to large errors (10). The techniques belonging to group (c) do not have the above limitations and, in principle, all three dynamic methods can be used to measure interfacial tension in aqueous/aqueous twophase systems; however, as the accuracy of the methods is similar (12), the drop retraction method was selected because as of its relative simplicity. In this method (13–15), the interfacial tension is calculated from measurements of the time evolution of the shape of an initially deformed drop after the applied stress or flow field is removed. It has already been shown that the video technique (16) enables good quality images of the dis-

ln D(t) = ln D0 −

t . τ

[5]

In this work the interfacial tension was calculated from Eq. [4] and Eq. [5]. 3. EXPERIMENTAL

3.1. Approach As phase separation occurs when aqueous solutions of two incompatible polymers are mixed together, in principle, it is possible to measure the interfacial tension in the dispersion obtained by direct mixing of a pure gelatin solution with a pure dextran solution. However, in such a case the volumetric ratio of the separated phases is not known a priori (unless an accurate phase diagram is known) and also the viscosity of separated phases is not known. Therefore, in the experiments reported below, a different procedure was adopted. First, aqueous solutions of both biopolymers were made (stock solutions). Next, those solutions were mixed, equilibrated, and separated and the density and viscosity of separated phases were measured. Finally, the interfacial tension between the gelatin-rich and the dextranrich phases was measured for mixtures that were made up from the previously equilibrated phases. 3.2. Materials The single biopolymer solutions of gelatin and dextran were prepared by dissolving dry gelatin (UG-719-N, Extraco, Sweden) and dextran powder (Dextran T2000, molecular weight 2,000,000, Amersham Pharmacia Biotech AB, Sweden) using an aqueous 0.1 M NaCl/0.05% sodium azide solution as a solvent. The pH was not controlled. The dextran solution was prepared by dissolving the powder at ambient temperature while stirring with a magnetic stirrer. The dry gelatin powder was first soaked for 30 min at ambient temperature and then dissolved at 60◦ C while stirring with a magnetic stirrer for 30 min. Biopolymers

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INTERFACIAL TENSION IN PHASE-SEPARATED MIXTURES

TABLE 1 Compositions of the Systems Prepared for Measurement of Interfacial Tension (See Also Fig. 1) 1

2

3

4

X G [% w/w] X D [% w/w]

0.7 4.8

1.0 6.8

1.5 10.2

7.0 6.8

Gelatin (wt%)

12

Mixtures

concentrations were chosen so that, for preparation of each of the four mixtures shown in Table 1, three parts of dextran solution were mixed with two parts of gelatin solution while mixing with a magnetic stirrer for 15 min at 60◦ C. The mixtures were equilibrated and separated (centrifugation, 9375g; temperature, 40◦ C) into gelatin-rich and dextran-rich phases. Concentrations of gelatin and dextran in the separated phases (gelatin-rich and dextran-rich) were measured using a combination of UV–visible spectroscopy and refractive index measurements, as described in detail below. Gelatin concentrations in each separated phase were measured using UV–visible spectroscopy. Calibration spectra were obtained for a series of standard gelatin solutions and these were used to determine the gelatin concentrations in each of the equilibrated phases by minimizing the function β=800nm  

A(ν) −

β=260nm

2 As (ν) · X G − a1 · ν 4 , Xs

XD =

dn d XD

,

B1

10 8

4 (6) A1

6 4

3 (5)

2

2 0 0

2

1

A2

4

6

B2 8

C2 10

12

Dextran (wt%) FIG. 1. Tie-lines at which the surface tension was measured (A1 A2 , B1 B2 , C1 C2 ) and points () representing the systems prepared for the interfacial tension measurements (1, 2, 3, and 4).

The effects on the interfacial tension of adding small amounts of low molecular weight dextran to the mixture were also investigated. To this end, two more systems were prepared by modifying systems 3 and 4. Dextran of molecular weight 10500 (Sigma Chemical Co., Lot 79H0631), 1% w/w, was added to the dextran-rich phase in system 3 (denoted as system 5), and dextran of molecular weight 39400 (Sigma Chemical Co., Lot 27H1213), 1% w/w, was added to the dextran-rich phase in system 4 (denoted as system 6). The interfacial tension between the gelatin-rich and the dextran-rich phases was measured for all six systems and the results are discussed in Section 4.

[6]

where A(ν) is the absorbence spectrum of the separated phase as a function of the wavelength ν, AS (ν) is that of a standard gelatin solution of weight concentration X S , and X G is the concentration of gelatin in the separated phase. The term a1 · ν 4 is used to correct for the scattering from the dextran in the separated phase. Both X G and a are treated as unknowns in the above. The concentrations of dextran X D in the separated phases were determined from refractive index measurements and calculated using the relation n meas − X G · ddn XG

C1

14

3.2.1. Rheology of Separated Phases The rheological properties of all the separated phases in systems 1 to 6 (see Fig. 1) were measured using a Carri-Med CSL rheometer. The rheology of systems (5) and (6) was identical to the rheology of systems (3) and (4) respectively; therefore the data for systems (5) and (6) are not shown. The experimental data for all other systems were fitted to a power law model,

[7]

µ = K γ˙ n−1 .

Where n meas is the refractive index of the separated phase and dn/d X was determined for gelatin and dextran using standard solutions. The technique was verified by measuring the concentrations of gelatin and dextran in aqueous mixtures of the two polymers with known concentrations. The results are shown in Table 2 and Fig 1.

[8]

and the results are summarized in Fig. 2 and Table 3. As expected, increasing the concentration of polymers leads to an increase in the viscosity of both separated phases. Both dextran-rich phases and gelatin-rich phases are very weakly non-Newtonian and in all cases but one n > 0.95. Such a high

TABLE 2 Concentrations of Gelatine and Dextran in Gelatin-Rich and Dextran-Rich Phases (Ends of Tie-Lines; See Fig. 1)

X G [% w/w] X D [% w/w]

A1

A2

B1

B2

C1

C2

5.42 ± 0.2 1.55 ± 0.4

0.53 ± 0.2 5.00 ± 0.4

9.10 ± 0.2 1.97 ± 0.4

0.56 ± 0.2 7.27 ± 0.4

13.65 ± 0.2 2.95 ± 0.4

0.59 ± 0.2 10.68 ± 0.4

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DING ET AL. 1

1

Viscosity [Pas]

Viscosity [Pas]

(b)

(a)

0.1

0.1

0.01

0.01 1

10

1

100

10

100 -1

-1

Shear rate [s ]

Shear rate [s ]

FIG. 2. Viscosity of separated phases at 40◦ C: (a) gelatin-rich phases, (b) dextran-rich phases, () 0.7% gelatin/4.8% dextran; () 1.0% gelatin/6.8% dextran; () 1.5% gelatin/10.2% dextran; () 7.0% gelatin/6.8% dextran.

value of n justifies the approximation that all the phases are Newtonian and the theory of small deformations (Eqs. [3], [4], and [5]), valid for Newtonian fluids, can be used to calculate the interfacial tension. 3.3. Experimental Apparatus and Procedure The experiments were carried out in a glass Couette device of height 100 mm (see Fig. 3) and with a gap between the cylinders of 2.5 mm (D0 = 85 mm, Di = 80 mm). The Couette device was placed in a water bath of square cross section made from optically flat glass to avoid distortion of the images of the drops and to control temperature to 40◦ C. A computer-controlled stepper motor drove the inner cylinder, and the outer cylinder was stationary. The drop sizes and the deformation parameters were measured using the video-microscope–computer system described in detail previously (16). The Couette device was initially filled with 70 ml of dextranrich continuous phase (the dextran-rich phase in systems 1, 2, 3, 5, and 6) and the liquid was stirred for a certain time to allow the temperature to equilibrate at 40◦ C. After that time 0.1 ml of gelatin-rich phase (dispersed phase) was added to the continuous phase using a syringe. The exception to this procedure was for system 4, where the phases were inverted: the gelatin-rich phase was used as the continuous phase and the dextran-rich phase as dispersed. The dispersion was stirred at a constant shear rate for

10 min and the deformation of the drops was recorded on videotape. After that time the motor was stopped, while the recording was continued for the duration of the drop retraction. The position of the drops during their retraction (retraction time 1.5 to 3 s) did not change significantly; therefore it can be assumed that the effect of buoyancy on drop deformation was negligible. A typical example of the images of the retracting drop after cessation of the shear flow is shown in Fig. 4. The images were digitized and the deformation parameter was measured as a function of time using image analysis software (16). It was possible to analyze up to 50 images of the retracting drop per second; however, it has been found that analyzing only 10 to 15 images gives sufficient accuracy. 4. RESULTS AND DISCUSSION

The results are presented in three sections. In Section 4.1, results of preliminary experiments, which were carried out to determine the range of experimental parameters required for interfacial tension measurements, are reported. In Section 4.2, the

3 8

TABLE 3 The Values of K and n for Gelatin-Rich Phases (G-r Phase) and Dextran-Rich Phases (D-r Phase) for All Investigated Systems System 1

K [Pasn ] n [−]

System 2

System 3

1

System 4

G-r phase

D-r phase

G-r phase

D-r phase

G-r phase

D-r phase

G-r phase

D-r phase

0.0377 0.915

0.013 0.968

0.047 0.982

0.024 0.987

0.158 0.971

0.052 0.980

0.109 0.990

0.063 1.00

7

5 4

6

2

FIG. 3. Experimental rig: 1—Couette device (Di = 80 mm, D0 = 85 mm), 2—water bath, 3—step motor, 4—computer, 5—stereo microscope fitted with 3CCD video camera, 6—video recorder, 7—monitor, 8—strobe light.

371

INTERFACIAL TENSION IN PHASE-SEPARATED MIXTURES

FIG. 4.

Examples of images of deformed drop retraction after cessation of shear flow.

effects on interfacial tension of the concentrations of gelatin and dextran and of the addition of small amounts of lowmolecular-weight dextran to the dextran-rich phase are discussed. In Section 4.3 the experimental results for the interfacial tension are compared with a theoretical prediction using a Flory–Huggins-based analysis of the measured tie-line data. 4.1. Selection of the Range of Experimental Parameters 4.1.1. Limits of Deformation Parameter Range To ensure high accuracy in the calculation of interfacial tension from measured values of the deformation parameters, these measurements must be taken within an upper and lower limit of the deformation parameter. The upper limit is determined by the main assumption of the small deformation theory (see Section 2, Eqs. [3], [4], and [5]). The lower limit arises from the experimental limitations; i.e., it is very difficult to measure accurately very small deformation parameters. Both limits were determined experimentally and typical results of the measurements of deformation parameter as a function of time are shown in Fig. 5. From Fig. 5 it is clear that the assumption of a linear relation between ln (D) and time is valid only for values of the deformation parameter smaller than 0.25, with r 2 = 0.9989; i.e., there

-0.5

D=0.28 D=0.25

ln(D) -1.0

is a nearly perfect fit of the experimental data to a straight line. Taking all the data points in Fig. 5 gives r 2 = 0.9861. The measurements of deformation parameters which were smaller than 0.03 leads to rather large scatter of experimental data (data not shown). Therefore, in the experiments reported in this paper the deformation parameter was measured from the time when the flow was stopped (highly deformed drops) until the time when the drop fully retracted, but in the calculation of interfacial tension, only values of the deformation parameter in the range 0.03 < D < 0.25 were used. 4.1.2. Selection of the Initial Shear Rate The initial shear rate (or flow field) should not affect the retraction process after the cessation of the flow; hence the calculated interfacial tension should be independent of the initial shear rate. However, the initial shear rate affects the drop size as well as the degree of deformation; therefore it might indirectly affect the accuracy of the measurements of the deformation parameter. To select the optimum shear rate for experiments, the effect of the shear rate on the accuracy of the calculated interfacial tension has been investigated using system 2 (separated 1% gelatin/6.8% dextran, Fig. 1) and the results are summarized in Fig. 6 and in Table 4. In Fig. 6a the logarithm of the deformation parameters of drops of different sizes (initial shear rate of 0.9 s−1 ) is plotted as a function of time, and in Fig. 6b the deformation parameters of drops of different sizes (initial shear rates 1.8 and 3.7 s−1 ) are plotted as a function of t/R0 . In both cases the experimental data fall on straight lines; however, the calculated interfacial

-1.5 D=0.03

D=0.34

TABLE 4 Interfacial Tension at Different Shear Rates for 1.0% Gelatin/6.8% Dextran, Gelatin-Rich Phase Dispersed

-2.0 -2.5

γ˙ = 0.9 s−1

-3.0 -3.5 0.0

0.5

1.0

1.5

2.0

2.5

Time [sec] FIG. 5. Deformation parameter as a function of time, gelatin-rich phase dispersed, dextran-rich phase continuous, both phases obtained from 1.0% gelatin/6.77% dextran system, R0 = 147 µm.

γ˙ = 1.8 s−1

γ˙ = 3.7 s−1

R0 (µm)

σ (µN/m)

R0 (µm)

σ (µN/m)

R0 (µm)

σ (µN/m)

253 223 170 —

8.77 8.35 10.30 —

147 109 165 —

9.49 11.20 10.36 —

131 109 126 91

9.89 9.81 10.28 10.77

σAv = 9.14 ± 1.03

σAv = 10.35 ± 0.85

σAv = 10.19 ± 1.44

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DING ET AL.

-1.0

0.50

ln(D)

D

(a)

-1.5 -2.0

0.10

-2.5

0.05

(b)

-3.0

t/R0

Time [sec] -3.5 0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.01 0.000

0.004

0.008

0.012

FIG. 6. Deformation parameter as a function of time for different initial shear rates in 1.0% gelatin/6.8% dextran system: (a) γ˙ = 0.9 s−1 , () R0 = 170 µm, () R0 = 223 µm, () R0 = 253 µm; (b) () γ˙ = 1.8 s−1 , R0 = 147 µm, () γ˙ = 1.8 s−1 , R0 = 109 µm, () γ˙ = 1.8 s−1 , R0 = 165 µm, () γ˙ = 3.7 s−1 , R0 = 131 µm, () γ˙ = 3.7 s−1 , R0 = 126 µm, () R0 = 97 µm.

tension at γ˙ = 1.8 s−1 has the narrower error band (see Table 4). Therefore, all experiments discussed below were carried out at this shear rate. 4.2. The Effect of Concentration of Biopolymers on Interfacial Tension The effect of biopolymer concentrations on the interfacial tension was investigated for two different cases. In the first case the change of concentrations of polymers was such that the new system stayed on the same tie-line (see Fig. 1, line C1 C2 , points 3 and 4). This meant that the concentrations of biopolymers in the separated phases were the same and only the average composition of the system (volume fraction of each phase) changed. In the second case, the change of concentrations was such that each new system was on a different tie-line (see Fig. 1, lines A1 A2 , B1 B2 , C1 C2 , points 1, 2, and 3). Therefore, both the concentration of biopolymers in separated phases and the volume fraction of the dispersed phase changed.

The experimental data (points in Fig. 8) were used to calculate the coefficients in Eqs. [1], [2], and [2a] using linear regression in log–log coordinates (solid lines) and the following best-fit equations were obtained: log σ = −1.3888 + 2.3524 · log(TLL) r 2 = 0.9988

[9]

log σ = −1.3429 + 2.4636 · log(X G ) r = 0.9962 [10] 2

log σ = −0.4777 + 2.0566 · log(X D ) r 2 = 0.9987. [11] Bearing in mind that there are only three points for the above analyses, limiting the degree to which one should interpret the results, nevertheless in all three cases there is a very good fit of the experimental data to the regression lines (r 2 > 0.99). These results show that any of the above three parameters can be used to correlate interfacial tension as a function of the composition of the phase-separated mixtures. Similar results were obtained by Forciniti (5) for the dextran/PEG system.

4.2.1. Interfacial Tension in the Systems at Different Tie-Lines The interfacial tension in each of three different systems representing three different tie-lines (Fig. 1, points 1, 2, 3) has been calculated from deformation parameters measured for at least three drops of different sizes. Examples of the raw data (deformation parameter as a function of time) are shown in Fig. 7. The experimental data fit the straight lines very well (r 2 > 0.99), which confirms that the small deformation theory used in this work to calculate interfacial tension can be used with full confidence. The interfacial tensions calculated from Eq. [4] for systems 1, 2, and 3 (see Fig. 1) are summarized in Table 5. As the concentration of biopolymers increases (the systems move away from the critical point) the interfacial tension also increases. In Fig. 8 interfacial tension is plotted as a function of the length of the tie-line (Fig. 8a), the difference of gelatin concentrations in the gelatin-rich and dextran-rich phases (Fig. 8b), and the difference of dextran concentrations in the dextran-rich phases and gelatin-rich phases (Fig. 8c).

-1.0

ln(D) -1.5 -2.0 -2.5 -3.0 -3.5 0.0

0.5

1.0

1.5

2.0

Time [sec] FIG. 7. Deformation parameter as a function of time measured with the phases obtained after centrifugation of 1.5% gelatin/10.2% dextran system; gelatin-rich phase dispersed shear rate γ˙ = 1.8 s−1 , () R0 = 79 µm, () R0 = 122 µm, () R0 = 159 µm, () R0 = 182 µm.

373

INTERFACIAL TENSION IN PHASE-SEPARATED MIXTURES

TABLE 5 Interfacial Tension at Different Tie-Lines; Separated Phases Were Obtained from the Mixtures Shown in the First Row 0.7% gelatin/ 4.8% dextran

1.0% gelatin/ 6.8%dextran

1.5% gelatin/ 10.2%dextran

R0 [µm]

σ [µN/m]

R0 [µm]

σ [µN/m]

R0 [µm]

σ [µN/m]

79 101 110 117

3.02 2.90 2.99 2.86

109 147 165 —

11.2 9.49 10.36 —

79 122 159 181

26.04 28.78 34.46 30.83

σAv = 2.94 ± 0.08

σAv = 10.35 ± 0.85

σAv = 30.02 ± 3.35

4.2.2. Interfacial Tension in Systems on the Same Tie-Line For the systems on the same tie-line two cases were investigated. In the first case the overall starting concentrations of the biopolymers were changed, so that different equilibrated phases were obtained (systems 3 and 4, Fig. 1). However, the structure of the dispersions used in the measurements was kept the same, with the gelatin-rich phase dispersed in the dextranrich phase. In both systems the interfacial tension was practically the same with σ = 30.02 ± 3.35 [µN/m] in system 3 (1.5% gelatin/10.2% dextran) and σ = 29.55 ± 3.39 [µN/m] in system 4 (7% gelatin/6.8% dextran). In the second case the starting concentrations of the system were the same (the same equilibrated phases were used) but the structure of dispersion was changed. Again, interfacial tension between the gelatin-rich drop and the dextran-rich continuous phase (σ = 29.55 ± 3.39 [µN/m]) was the same as interfacial tension between the dextran-rich drop and the gelatin-rich continuous phase (σ = 29.01 ± 0.3 [µN/m]). Within experimental error the interfacial tension for all the systems on the same tie-line is practically constant and independent of the starting concentration of the polymers and of the structure of the dispersion. This can be explained by the fact that the compositions of the gelatin-rich phase and the dextranrich phase are similar, so that physical properties, including the interfacial tension, should also be the same.

50

4.2.3. Effect of Addition of Low Molecular Weight Dextran to Dextran-Rich Phase on Interfacial Tension In order to check the sensitivity of interfacial tension to the presence of smaller dextran molecules in the system, a small amount (1% w/w) of low molecular weight dextran was added to the dextran-rich phase. Two types of low molecular weight dextran were used: 10,500-MW dextran was added to the dextranrich phase obtained from system (3) and 39,400-MW dextran was added to the dextran-rich phase obtained from the system (4) (see Fig 1). The measured interfacial tension was compared with the “pure” systems and the results are summarized in Table 6. In both systems the presence of small amounts of low molecular weight dextran appears to slightly decrease the interfacial tension (compare the average values in Table 6). However, the differences between the interfacial tension in the “pure” systems and in the systems containing low-molecular-weight dextran are within the range of experimental error. It is possible that the effect is small because of the broad molecular weight distribution of the dextran we used. This may mean that there is a substantial amount of low molecular weight material already present in the mixtures, so adding more has little effect. More experimental data are necessary to fully access this effect. 4.3. Interfacial Tension Calculations for Phase-Separated Gelatin–Dextran Mixtures In addition to the correlations based on tie-line length, discussed above, interfacial tensions were calculated from experimental tie-line data for gelatin–dextran solutions using a self-consistent field computer model for ternary systems, first described by Hong and Noolandi (18, 19). In this calculation, the Flory–Huggins theory is assumed to describe the solution behavior, and interfacial properties are calculated by free energy minimization. When this is done, a compromise in the interfacial width is sought between the reduced amount of the interfacial region achieved by its being narrow, and the reduction in concentration gradients achieved by its being wide. In the calculations, the polymer chains are idealized as random walks, each walk occurring in the mean field created by all the

50

σ [µN/m]

30

50

(a)

σ [µN/m]

30

10

(b)

10

∆ XG [%w/w]

∆ XD [% w/w]

2 10

15

20

(c)

10

TLL [% w/w] 2 5

σ [µN/m]

30

2 5

10

15

2

3

5

10

FIG. 8. Interfacial tension as a function of (a) length of tie-line, (b) difference in gelatin concentration in gelatin-rich and dextran-rich phases, (c) difference in dextran concentration in dextran-rich and gelatin-rich phases.

374

DING ET AL.

TABLE 6 Effect of Addition of Low Molecular Weight Dextran to the Dextran-Rich Phase on Interfacial Tension 7.0% gelatin/6.8% dextran R0 [µm]

σ [µN/m]

R0 [µm]

99 137 159 —

26.16 29.93 32.56 —

115 138 164 —

σAv = 29.55 ± 3.39 a b

a

1.5% gelatin/10.2% dextran σa

[µN/m]

R0 [µm]

σ [µN/m]

R0 b [µm]

σ b [µN/m]

79 122 159 181

26.04 28.78 34.46 30.83

116 168 187 —

29.59 29.27 27.52 —

27.1 28.94 27.51 —

σAv = 27.85 ± 1.09

σAv = 30.02 ± 3.55

σAv = 28.79 ± 1.27

1% w/w of 10,500-MW dextran added to dextran-rich phase in 7% gelatin/6.8% dextran system. 1% w/w of 39,400-MW dextran added to dextran-rich phase in 1.5% gelatin/10.2% dextran system.

other polymer segments and solvent molecules in the system. These walks are described by two coupled partial differential equations, one for each polymer type. The solutions of these equations allow the equilibrium polymer segment (and solvent) densities to be calculated across the interface. Obvious boundary conditions are that these densities must have the bulk values predicted by the Flory–Huggins theory at long distances from the interface. Since the diffusion equations explicitly contain the field, or segment free energy, appropriate to a given point within the interface, and this depends on the segment densities being sought, an iterative approach (18) is required for their solution. An equilibrium surface tension is then calculated from the final, converged, segment density functions as an excess free energy per unit area (19). Application of this approach to biopolymer mixtures will be described in more detail elsewhere. Here, results for the tielines measured for gelatin–dextran will be summarized. First, the three experimental tie-line concentrations (Fig. 1, Table 2) had to be converted to volume fraction units using partial specific volumes of 0.71 and 0.61 ml/gm for gelatin and dextran, respectively. Each tie-line was then fitted precisely using the Flory–Huggins theory adapted to a ternary system. This has been described in detail elsewhere (20) and involved selecting fixed values for the gelatin molecular weight (Mn = 67,572, from experiment) and the gelatin–solvent  value (0.485 taken as a literature average). The fit then uniquely determined the dextran (number average) molecular weight, the dextran–solvent  value, and the gelatin–dextran interactive  value. Separate values were obtained for these three parameters for each tieline. Ideally, these should have been identical, but in practice, when this tie-line fitting procedure was attempted, a spread of results was obtained, including rather unsatisfactory results for the dextran–solvent  value (experimental estimates are available from virial coefficient data). This discrepancy was particularly obvious for the two more concentrated tie-line mixtures and forced an adjustment to be made of the corresponding fixed gelatin–solvent  parameter. A value of 0.499 was found to be more acceptable, suggesting that the gelatin was much closer to the θ condition than anticipated, this possibly being related to pH. A second unsatisfactory feature of the fits was also noted.

The calculated dextran molecular weight also varied with tie-line and, on average, was much lower than suggested by experiment. This could be a result of the high molecular-weight polydispersity of the dextran (not included in the calculation) and was ignored in the surface tension computations. Calculated interfacial tensions for the Flory–Huggins tie-line fits using the best choice of the gelatin–solvent χ parameter are compared with experiment in Table 7. The agreement for the most dilute mixture is remarkably good considering the problems with tie-line description just discussed. It should be noted that no extra parameter is introduced when the surface tension is calculated, so this is essentially an add-on prediction of the Flory–Huggins theory. Agreement is less satisfactory for the remaining two tie-lines, however, particularly for the most concentrated system. This seems to coincide with the rather poor result for the dextran–solvent  parameter also obtained from this tie-line, even after the gelatin  value is modified. Evidently, doubt attaches to the accuracy of tie-line measurement, or to the capacity of the Flory–Huggins approach to describe it, or to a combination of these. To illustrate how the surface tension might change if errors were present in the two more concentrated experimental tielines, the Flory–Huggins parameters derived from the most dilute example were used to make theoretical predictions about the forms the two additional tie-lines should have if the Flory– Huggins model were followed rigorously. The effect was to lengthen these two tie-lines toward the gelatin axis and to rotate them to make their intercepts on the concentration axes more equal. New estimates of surface tension were then calculated, TABLE 7 Comparison of Measured Interfacial Tensions (σAV ) for Three Tie-Lines with Calculated Values Using Experimental (1) and Hypothetical (2) Tie-Lines Tie-line

σAv (experiment) [µN/m]

σ (calculation 1) [µN/m]

σ (calculation 2) [µN/m]

1 2 3

2.94 ± 0.08 10.35 ± 0.85 30.02 ± 3.55

2.8 7.3 13.4

2.8 15.8 42.4

INTERFACIAL TENSION IN PHASE-SEPARATED MIXTURES

the results also appearing in Table 7. These new calculated values are now higher than experiment, agreement being slightly worse for tie-line two, but significantly better for tie-line three. In fact, the average of the two sets of theoretical values in Table 7 is very close to experiment. It seems likely that some refinement of the experimental tie-line data is necessary, but perhaps not as much as is suggested by following the Flory–Huggins predictions of the first tie line exactly. Finally, it is of interest that the second set of calculated tie-lines based on a constant set of Flory–Huggins parameters supports the experimental conclusion of this paper that surface tension varies quite strongly with tie-line length. According to the Hong and Noolandi (18, 19) model, the surface tension has a value of zero at the critical point, but increases according to a power law of roughly 2.7 with tie-line length away from this point. The experimental power law suggested here is slightly lower than this at 2.2. It is concluded that, despite the gross approximations of the Flory–Huggins approach, particularly when treating polymers as complex as biopolymers, and water as a solvent, the surface tension characteristics of water-in-water emulsions are described realistically and to an extent quantitatively. A conclusive demonstration of this would, however, require more accurate tie-line measurements and data for more than three tielines. 5. CONCLUSIONS

The modified retractive drop method has been applied to the measurement of interfacial tension in phase-separated gelatin/ dextran systems. Despite relatively high viscosity and very low density difference between the phases, this method allows accurate measurement of interfacial tension without withdrawing the samples from the dispersion. The results show that interfacial tension depends on the concentration of biopolymers and increases as distance from the critical point increases. However, it is constant for the same tieline and it is independent of the viscosity ratio λ, of the curvature of the interface, and of the structure of dispersion (e.g., which phase is dispersed and which continuous). The same values of interfacial tension were also obtained from different-sized drops, confirming that the employed technique is accurate. The interfacial tension is equally well correlated with the length of the tie-line and with the difference in concentration of each polymer in the separated phases. Taking into account that the interfacial tension is constant at the same tie-line (at different concentrations of polymers) it appears that correlating interfacial tension with the length of the tie-line is less ambiguous than correlating with concentration difference. Preliminary investigations of the effect of low molecular weight of dextran on interfacial tension at the same tie-line show that addition of low-molecular-weight dextran to a dextran-rich phase slightly reduces the interfacial tension. The analysis of the interfacial tension as a function of the tieline length can be taken further. Using a Flory–Huggins-based

375

theoretical approach, it is possible to make predictions of the interfacial tension, based on the measured tie-line data. Despite the gross approximations inherent in this approach, particularly when treating polymers as complex as biopolymers, and water as a solvent, the surface tension characteristics of water-inwater emulsions are described realistically and to an extent quantitatively. APPENDIX: NOMENCLATURE

a b D D0 K n R0 t t0 TLL XD XG λ µc µd σ σAv τ X D X G

longer axis of deformed drop shorter axis of deformed drop deformation parameter = (a − b)/(a + b) deformation parameter at t0 consistency constant power-law index radius of fully retracted drop time time at which drop retraction measurement starts  length of tie-line, (X D )2 + (X G )2 concentration of dextran concentration of gelatin viscosity ratio (µd /µc ) viscosity of continuous phase viscosity of dispersed phase interfacial tension arithmetic average of measured interfacial tension dimensionless time Eq. [3] difference between dextran concentrations in separated phases difference between gelatin concentrations in separated phases

[m] [m] [—] [—] [Pasn ] [—] [m] [s] [s] [% w/w] [% w/w] [% w/w] [—] [Pas] [Pas] [µN/m] [µN/m] [—] [% w/w] [% w/w]

ACKNOWLEDGMENTS One of us (P. Ding) acknowledges financial support from Unilever Research Labs Sharnbrook.

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