Nanoscale fluctuations and surface tension measurements in droplets using phase-resolved low-coherence interferometry

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1 Nanoscale fluctuations and surface tension measurements in drolets using hase-resolved low-coherence interferometry Ru Wang, 1 Taewoo Kim, 2 Mustafa Mir, 2 and Gabriel Poescu 2, * 1 Quantitative Light Imaging Laboratory, Deartment of Mechanical Science and Engineering, Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Chamaign, Urbana, Illinois 61801, USA 2 Quantitative Light Imaging Laboratory, Deartment of Electrical and Comuter Engineering, Beckman Institute for Advanced Science and Technology, University of Illinois at Urbana-Chamaign, Urbana, Illinois 61801, USA *Corresonding author: goescu@illinois.edu Received 16 August 2012; revised 15 October 2012; acceted 16 October 2012; osted 16 October 2012 (Doc. ID ); ublished 13 November 2012 A common-ath interferometer was designed with raidly tunable broadband swet laser source, which rovides quantitative hase measurements of nanometer scale motions with very high sensitivity. With this setu, we are able to detect the thermal fluctuations in liquid drolets hanging at the end of an otical fiber. The measured nanoscale dislacement fluctuations of various drolet surfaces were used to extract the surface tension. This newly develoed technique roved the feasibility of noninvasive, fast, hase-resolved dynamic light scattering measurement of fluid mechanical roerties Otical Society of America OCIS codes: , , Introduction Soft materials, such as olymer solutions, colloids, and biological matter, are characterized by multile characteristic satiotemoral scales and, as a consequence, their dynamic resonse to external strains is nontrivial [1]. Microrheology studies the mechanical resonse of such materials at the microscoic scale. This tye of investigation is relevant to a number of fields, esecially biology, where cell mechanics is essential for the roer functioning of a living system [2]. It has been shown that otical methods (e.g., dynamic light scattering, fluorescence correlation sectroscoy, otical traing) are uniquely suitable for studying dynamic roerties of soft matter [3]. Recent microrheology techniques rely on alying X/13/01A177-05$15.00/ Otical Society of America strain to the fluid through embedded robe articles, either undergoing thermal (Brownian) or externally driven motion [4,5]. In such techniques, the statistics of article motion is described via the ensembleaveraged mean squared dislacements, which requires the recording of multile trajectories. Here we resent measurements of the nanoscale surface fluctuations associated with small fluid drolets at thermal equilibrium. Using the fluctuationdissiation theorem [6], these motions are used to infer the mechanical roerties of the fluid. Our exerimental setu is imlemented with otical fibers and relies on hase-resolved interferometry with broadband fields. Amlitude-based low-coherence interferometry (LCI) has been used in the ast for erforming dynamic light scattering and microrheology measurement [7 11]. However, such amlitudebased techniques are not sensitive enough to detect quantitatively nanometer scale motions; 1 January 2013 / Vol. 52, No. 1 / APPLIED OPTICS A177

2 quantitative hase measurements are necessary instead [12,13]. We show that resolving the hase of the field allows us to quantify these fluctuations and extract the resective surface tension coefficient of the liquid. 2. Exerimental System Our exerimental system is described in Fig. 1. Light from a swet source, i.e., a laser whose emission wavelength is sweeing a 100 nm sectral range around λ nm at frequency of 8 khz, is couled into a single mode fiber couler. A drolet of interest is in contact with the end of the single-mode otical fiber such that it is susended in air. The otical field emerging from the fiber core is artially secularly reflected at the fiber/fluid interface, while the transmitted comonent roagates through the drolet, reflects at the fluid air interface, and roagates back into the fiber. Thus, the secular field, U 1, can be considered as a reference for the samle field, U 2, which carries information about the osition of the drolet surface. The two fields roagate back through the outut of the 1 2 fiber couler and to the detector, which is a conventional OCT balanced detector. The intensity measured by the detector at each otical frequency ω has the form I ω ju 1 ω U 2 ω j 2 I 1 I 2 2 Re W 12 ω : (1) In Eq. (1), I 1;2 are the intensities of the two fields and W 12 is the cross-sectral density associated with the two fields, which is defined as [14] W 12 ω U 1 ω U 2 ω ju 1 ω jju 2 ω je i ωτ0 φ : (2) In Eq. (2), τ 0 2nd c, where n is the refractive index of drolet and d is the thickness of the drolet, for the case of water n τ 0 indicates the average time delay associated with U 2 due to roagation through the drolet and φ is the hase shift due to the drolet fluctuations around the average thickness. Note that the two fields are identical in terms of frequency content, excet for the hase term and irrelevant scaling of the amlitude. Thus, the modulus of the cross sectral density is the ower sectrum of the original field, S ω ju 1 ω j 2, and Eq. (2) may be written as W 12 ω S ω e i ωτ 0 φ : (3) From Eq. (1), it is clear that exerimentally we only have access to the real art of the cross-sectral density. However, the imaginary art of W is obtained using a Hilbert transform as Im W 12 ω 1 π P Z Re W 12 ω 0 ω ω 0 dω 0 ; (4) where P indicates a rincial value integral. From W 12, we can now exress the cross-correlation function, Γ 12, via a Fourier transform Γ 12 τ Z Combining Eqs. (3) and (5), we have W 12 ω e iωτ dω: (5) Γ 12 τ Γ 0 τ τ 0 e iφ ; (6) Fig. 1. (Color online) Phase-resolved LCI setu and the exanded view of the fiber ti. The Gaussian field roagation inside the drolet is shown. where Γ 0 is the autocorrelation function of either field, the Fourier transform of which is the sectrum, S. Thus, Eq. (6) indicates that the thickness of the drolet can be obtained from this shift as d cτ 0 2n. The resolution in measuring this thickness is given by the modulus of Γ 0, i.e., the coherence length of the field, which is 17.6 μm in our case. However, the fluctuations of the free surface can be measured with much better accuracy, as this information is carried by the hase of the cross-correlation. Here, it is imortant to note that, because the two interfering fields are traveling along a common ath, the hase relationshi between them is extremely stable. The standard deviation of the hase noise was measured by assessing the hase noise in the absence of a samle. The resulting athlength noise level was measured as 0.62 nm, which sets the limit on the lowest athlength fluctuation that the instrument can detect. This value translates to a nm change in thickness associated with the drolet. A178 APPLIED OPTICS / Vol. 52, No. 1 / 1 January 2013

3 3. Princile and Exerimental Validation Our system erforms 8,000 scans er second. Figure 2(a) shows an examle of time-resolved athlength distributions obtained by measuring a water drolet. In Fig. 2(b), we show a athlength rofile at aroximately t 30 ms, indicated by an arrow in Fig. 2(a). The eak associated with the reflection from the drolet is clearly observed at μm. The resective hase measurement is shown in Fig. 2(c). Outside the eak of interest, there are areas where the hase ma suffers from hase wraing due to lack of signal. However, in order to resolve the drolet fluctuations, only the hase information around the reflection eak is necessary. Such timeresolved hase traces are shown in Fig. 2(d) for alcohol, water, and oil drolets. Interestingly, the decay in time of the athlength signal is due to the evaoration of the drolets. As exected, the alcohol evaorates at the fastest rate, i.e., 8.5 μm s, and the oil evaoration was measured to be 100 times slower. To study the drolet fluctuations, we first fit the linear sloe and subtract it from the data. The inset of Fig. 2(c) shows the fluctuation track after subtracting the evaoration contribution. The ower sectra of these fluctuations for water, alcohol, and oil drolets are shown in Figs. 3(a) 3(c). Interestingly, all the ower sectra for these liquids exhibit a ower law behavior, with an exonent of α 2. We fit these data with a model borrowed from light scattering by liquid surfaces [15]. For a flat surface of a liquid at thermal equilibrium, the dislacement, h has a q deendence on the in-lane wavenumber of k k 2 x k 2 y, and vibrating frequency Ω 1 iωt h k ; Ω iΩt 0 iω hy 1 iωt 0 2 i ; (7) 1 2iΩt 0 where y σρ 4η 2 k, t 0 ρ 2ηk 2,andρ, η, and σ reresent the density, viscosity, and surface tension, resectively. Note that the Gaussian field incident on the drolet surface has its own satial deendence due to the original field at the end of the fiber and the roagation through the fiber. Thus, the calculation of the field of interest U 2 [Eq. (1)] is erformed in several stes. First, the Gaussian field exiting the fiber, U 0 r e r2 w 2 0, (where w 0 is the beam waist and r x 2 y 2 is the radial coordinate) roagates through the drolet to yield field U 0 0 ρ0 at the surface. Uon reflection off the fluctuating surface, the field becomes U 00 0 ρ0 ; Ω U 0 0 ρ0 e i2k 0h ρ 0 ;Ω, where k 0 is the light mean wavenumber and ρ 0 is the conjugate variable to k. After Gaussian roagation back to the fiber, we obtain the final field U 2.Of course, the calculation is erformed more efficiently in the k variable, as in this case the Fresnel roagation of Gaussian beams reduces to a roduct with a quadratic hase term. The final field U 2 can be exressed symbolically as a function of the initialfield U 0 and dislacement distribution h: (a) (b) (c) (d) Fig. 2. (Color online) Result of a hase resolved LCI measurement on a liquid drolet. (a) Modulus of the cross-correlation function measured over 100 ms. (b) Vertical rofile taken from (a). The arrow indicates the drolet thickness of μm. (c) Phase of the cross-correlation function over 100 ms; inset: Horizontal rofile at the arrow after removing the sloe. (d) Horizontal rofile taken from (c) for ethanol, water, and the immersion oil. The evaoration seed calculated from the sloe of each rofile is shown. 1 January 2013 / Vol. 52, No. 1 / APPLIED OPTICS A179

4 Fig. 3. (Color online) Power sectra of the surface fluctuation (left) and the histogram of the fluctuation amlitude distribution (right) for (a) ethanol, (b) water, and (c) immersion oil. The red lines in the ower sectra show the calculated result from our model and the red lines in the histogram lots show Gaussian fits (full width half maxima as indicated). U 2 k ;Ω ex i2k 0 h k ;Ω h z i U 0 k ex i k 2 ex i d k 2 : 2k 0 2k 0 (8) In Eq. (8), symbol indicates a 2D convolution integral with resect to variable (K x, K y )andd is the thickness of the drolet. Physically, this convolution indicates averaging of the high frequency riles in the surface (of high k values). The two roducts by ex i d 2k 0 k 2 indicate Gaussian roagation back and forth through the drolet; lane wave terms, ex ik 0 d, were ignored as they do not contribute to the fluctuation sectrum. Finally, since the otical fiber integrates the information satially, A180 APPLIED OPTICS / Vol. 52, No. 1 / 1 January 2013

5 Table 1. Physical Constants of Different Liquids n ρ (kg m 3 ) η (mns m 2 ) σ (mn m) Ethanol Water Oil the field U 2 is integrated with resect to k from 0 to 10;000 rad m which corresonds to about 1 μm wavelength. The numbers used for this calculation for each liquid is shown in Table 1. All the hysical constants are given at room temerature 25 C and the test oil samle was transarent oil, which is widely used for immersion oil objective to increase the resolution of a microscoe. The fluctuating hase of U 2 is extracted as shown in Eq. (6) and its ower sectrum is used to fit the exerimental data. As indicated by the high quality fit (r ), this theoretical model describes the data very well. From the integral of the ower sectrum, which is essentially the variance of the fluctuation, the surface tension of ethanol aqueous solutions in different concentrations is measured. As shown in Fig. 4, the measured data agree with the exected values within the exerimental errors [16]. The error bars indicate variability from measurement to measurement (N 10 for each concentration). Our errors are surrisingly high desite the sensitivity of our measurement and their understanding requires further investigation. The major ossibility could be that the surface fluctuation of drolet is very small (nm scale), so the environmental air flow around it could greatly change its rofile, which might contribute large errors. And the laser may increase the local temerature which might result in some variation of hysical constants we used to calculate the exected value. Surface Tension (mn/m) Exected Measured Alcohol Mole Fraction (%) Fig. 4. (Color online) Estimated surface tension of water ethanol mixture at various mole concentrations (red circles) and exected value of water ethanol mixture surface tension (black squares). Error bars were obtained from 10 different measurements made for each concentration. 4. Summary and Discussion In summary, we have introduced a measurement method for nanoscale surface thermal fluctuations of liquid drolets using hase-resolved low coherence interferometry. With the aid of low coherence fields, common-ath interferometry, and a fast samling rate, the sensitivity to thickness fluctuations is a fraction of a nanometer. The data is modeled based on the fluctuations-dissiation theorem to yield the surface tension of liquid drolets. The measurement of ethanol aqueous solutions at different concentrations shows good agreement with the exected hysical values. We anticiate that such a fiber otic sensor will find imortant alications in studying the mechanical roerties of small quantities of fluids, e.g., cytoskeleton solutions or blood lasma, without the need for robing beads. This research was suorted by National Science Foundation (CBET CAREER, CBET MRI). More information can be found at htt://light.ece.illinois.edu/. References 1. P. M. Chaikin and T. C. 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