INSTRUMENT AND METHOD FOR ACCURATE MEASUREMENT OF SURFACE VISCOSITY OF VISCOUS LIQUID

Abstract

A method is provided for determining the surface viscosity of a liquid in which a thread is formed from a drop of the liquid. The thread is lengthened and its minimum radius h.sub.0 is determined at multiple times between the thread formation and thread pinch-off. The minimum radius and associated time values are used to determine a linear relationship of minimum radius and time, with the coefficient of the linear relationship, or the slope X of the line in the linear relationship, corresponding to the surface viscosity μ.sub.s of the liquid according to one of the following equations:

[00001]$\begin{array}{cc}x=\frac{0.0709}{1+5B_{\mathrm{s0}/3h_0}},& \left(1\right)\end{array}$

where B.sub.s0=μ.sub.s/μR in which h.sub.0 is defined as above, R is the dimension of the feature on which the drop is provided and μ is the bulk viscosity of the liquid, or

[00002]$\begin{array}{cc}x=\frac{0.0304}{\mathrm{Oh}\left(1+5b_{\mathrm{s0}/3h_0}\right)},& \left(2\right)\end{array}$

in which Oh=μ/√{square root over (ρRσ)}, where μ and R are as defined above, ρ is the density of the liquid, and σ is the surface tension of the liquid without surfactants.

Claims

1. A method for determining the surface viscosity of a liquid comprising: providing a drop of a liquid for measuring the surface viscosity; forming a thread from the drop; at a start time, determining an initial radius h.sub.0 of the thread; increasing the length of the thread from the start time until the thread pinches off at an end time; at discrete time intervals between said start time and said end time, determining the minimum radius h.sub.min of the thread; and storing, in a computer memory, the minimum radius h.sub.min and the particular time interval values; after the end time, using the stored minimum radius and time interval values, determining a linear relationship between the minimum radius h.sub.min and time; determining a slope X of the line in the linear relationship; calculating, from the slope X, the surface viscosity μ.sub.s for the liquid according to one of the following equations; $\begin{array}{cc}x=\frac{0.0709}{1+5B_{\mathrm{s0}/3h_0}},& \left(1\right)\end{array}$ where B.sub.s0=μ.sub.s/μR in which h.sub.0 is defined as above, R is the dimension of the feature on which the drop is provided and μ is the bulk viscosity of the liquid, or $\begin{array}{cc}x=\frac{0.0304}{\mathrm{Oh}\left(1+5B_{\mathrm{s0}/3h_0}\right)},& \left(2\right)\end{array}$ in which Oh=μ/√{square root over (ρRσ)}, where μ and R are as defined above, ρ is the density of the liquid, and σ is the surface tension of the liquid without surfactants; and then storing the value for the surface viscosity μ.sub.s for the liquid in a memory.

2. The method of claim 1, wherein the liquid includes a surfactant.

3. The method of claim 1, wherein equation (1) is selected when the Reynolds number for the liquid is small, and equation (2) is selected when the Reynolds number for the liquid is large.

4. The method of claim 3, wherein equation (1) is selected when the Reynolds number for the liquid is 1 or less and equation (2) is selected when the Reynolds number for the liquid is greater than 1.

5. The method of claim 1, wherein the step of determining the minimum radius of the thread includes: digitizing the photographic image; detecting the edges of the thread in the digitized image; measuring the pixel width of the thread between the edges along the length of the thread; and identifying the minimum pixel width as the minimum radius of the thread for the digitized image.

6. The method of claim 1, further comprising: obtaining a photographic image of the drop; determining, from the photographic image, the dimension R.

7. The method of claim 1, wherein the step of providing a drop includes: supplying the liquid to a nozzle; and creating a drop from said nozzle, wherein the dimension R is the radius of the nozzle.

8. The method of claim 1, wherein the step of providing a drop includes positioning a drop between the ends of two rods, wherein the dimension R is the radius of the rods.

9. The method of claim 1, wherein the steps of determining the initial radius h.sub.0 of the thread and determining the minimum radius h.sub.min of the thread include: obtaining a photographic image of the thread; and determining the initial radius and minimum radius from the photographic image

Description

DESCRIPTION OF THE DRAWINGS

[0015] The above and other objects, features and advantages of various embodiments of the present disclosure will become more apparent when taken in conjunction with the following description and drawings, wherein identical reference numerals have been used, where appropriate, to designate identical features that are common to the figures, and wherein:

[0016] FIG. 1 are enlarged images of a thread after pinch-off showing the formation of unwanted satellite droplets.

[0017] FIG. 2 are diagrams comparing the effect of surfactants on drop and thread formation of a liquid.

[0018] FIG. 3 are images of the growth of a thread until pinch-off and the formation of satellite droplets thereafter.

[0019] FIG. 4 is a diagram of a drop and the initial formation of a thread of a liquid.

[0020] FIGS. 5A-D are images of the formation of a drop and thread of a liquid.

[0021] FIG. 6 is a graph of the minimum thread radius of a liquid thread as a function of time.

[0022] FIG. 7 is schematic of a system for determining the surface viscosity of a liquid according to one embodiment of the disclosure.

[0023] FIGS. 8A-8B form a flow chart of steps of a method for determining the surface viscosity of a liquid.

[0024] FIG. 9 is a flow chart of steps of another method for determining the surface viscosity of a liquid.

DETAILED DESCRIPTION

[0025] The present disclosure contemplates a system and method for accurately measuring surface viscosity of a liquid during drop and thread formation, whether or not the liquid includes a surfactant. The method for measuring surface viscosity is based on a discovery by the present inventors that the surface viscosity is related to the change in thread radius over time. As shown in FIG. 4, a drop starts from liquid exiting the nozzle at a nozzle radius R. As the thread forms it assumes a minimum radius, h.sub.min, at the drop-thread interface. Over time, as the drop begins to fall under the force of gravity, the minimum radius of the thread decreases, as depicted in FIGS. 5a-d, until the minimum radius is zero—the point at which the thread pinches-off from the drop. In particular, it was discovered that the surface viscosity of the liquid is related to the slope of the change in the minimum thread radius h.sub.min from the initial formation of the thread to the pinch-off point. With respect to FIG. 5, the change in minimum thread radius is measured from a time at the formation of the thread when the thread is at an initial minimum thread radius h.sub.0 depicted in FIG. 5b, and the time at which the drop separates from the thread, as depicted in FIG. 5d.

[0026] It can be appreciated that the approach shown in FIGS. 5a-d is only one technique to thread formation. In another technique, a liquid drop is positioned between the end of two rods having the radius R. The thread is formed as the rods are moved apart. Other techniques for forming a liquid drop and thread are contemplated, with the understanding that the dimension of the feature on which the drop is formed has the radius R.

[0027] More specifically, it has been discovered that the change in minimum thread radius, for a low Reynolds number liquid with a surfactant, is given by the equation:

[00005]$\begin{array}{cc}{h}_{\min }=\frac{0.0709}{1+5B_{s0/3h_0}}\tau ,& \left(1\right)\end{array}$

where B.sub.s0=μ.sub.s/μR in which μ.sub.s is the surface viscosity for the liquid at the time of thread formation, μ is the bulk viscosity of the liquid and R is the dimension of the feature on which the drop is formed, such as the nozzle radius.

[0028] For a liquid with a surfactant having a large Reynolds number the equation becomes:

[00006]$\begin{array}{cc}{h}_{\min }=\frac{0.0304}{Oh\left(1+5B_{\mathrm{s0}/3h_0}\right)}\tau .& \left(2\right)\end{array}$

[0029] In Equation (2), Oh is the Ohnesorge number, or the inverse of the Reynolds number, which is given by the equation Oh=μ/√{square root over (ρRσ)}, where μ and R are defined above, ρ is the density, and σ is the surface tension of the liquid without surfactants. Equation 2 is preferably used when the Reynolds number for the liquid is greater than 1.

[0030] The development of these equations is described in detail in Appendices B and C, respectively, which disclosures are expressly incorporated herein by reference. As can be appreciated from the form of the two equations, the coefficient of τ is the slope X of the change in h.sub.min as a function of time τ, as reflected in the graph of FIG. 6. (In the graph, the time τ is measured “backwards” from the time when the thread is pinched off to the time when the thread starts to form.) If the slope is known, it is a matter of simple arithmetic to calculate the unknown constant B.sub.s0 from which the surface viscosity μ.sub.s for the liquid can be calculated.

[0031] The present disclosure contemplates a measurement system that can determine the slope X, or the coefficient, of Equations (1) and (2). In one embodiment, the measurement system 10, shown in FIG. 7, includes a nozzle 11 for producing a drop of a liquid and a digital camera 12 that captures images of the drop and thread formation, as shown in FIG. 5. In one configuration, the nozzle is supplied with the liquid from a source, such as a syringe pump 13, that can be operated in a controlled manner to provide the liquid to the nozzle at a desired pressure and flow rate to produce a single drop and following thread. It can be appreciated that other means for forming the drop and thread can be employed, provided the drop and thread can be visualized by the camera 12.

[0032] The camera acquires an image of the drop as it falls and extends the length of the thread at multiple times T. A light source 14 can be provided to illuminate or back-light the drop as it falls and as the thread elongates, thins and eventually breaks. The camera continues to acquire images until the thread pinches off—i.e., until the radius of the thread reaches zero.

[0033] The multiple images acquired by the camera are provided to an image processor implemented within a computer 15. In one embodiment, the camera can be configured to obtain discrete images at fixed time intervals. In an alternative embodiment, the camera can be a movie camera and the image processor can be configured to capture discrete images at the fixed time intervals. In either embodiment, the image processor is provided with multiple discrete images of the drop and, more importantly the thread, from the creation of the thread to the pinch off of the thread. The multiple images can be digital or subsequently digitized using well-known digitization methods or software. The image processor is configured to determine the nozzle diameter, 2R, the initial minimum width or diameter of the thread, 2h.sub.0, and the minimum width or diameter of the thread, 2h.sub.min, at each time interval from the digitized images. The image processor can incorporate edge detection methods or software, as is well-known in the art, to determine the edges of the drop and especially the thread. The processor can then determine the pixel width between the detected edges along the vertical length of the drop and thread. The smallest pixel width along the vertical length is identified and that pixel width and the particular time interval is stored for further processing. This process is repeated over several time intervals, with the object being to accumulate (h.sub.min, τ) pairs sufficient to accurately establish the relationship between the minimum thread radius and time. As reflected in the graph of FIG. 6 that relationship is substantially linear. It can be appreciated that the accuracy of the relationship is a function of the number of (h.sub.min, τ) data pairs that are used to generate the line shown in FIG. 6. In one embodiment, 100 images are acquired from the creation of the thread to its break-up.

[0034] The computer 15 can process the stored h.sub.min vs. τ data to determine the slope X of the line. As needed, the computer can be configured to linearize the stored data, using best-fit or curve-fitting software as is known in the art, to extrapolate the line shown in the graph of FIG. 6, from which the slope X can then be determined. The computer is then configured to calculate the value for B.sub.s0 according to the equations (1) and (2), and then calculate the surface viscosity μ.sub.s of the liquid.

[0035] The system 10 shown in FIG. 7 is configured to obtain measurements for the surface viscosity of a liquid according to the method steps in FIG. 8. In the first Step 101, liquid is supplied from the source 13 to the nozzle 11 at a pressure and flow rate sufficient to form a single drop at the nozzle. Once the drop is formed, the camera 12 is operated to obtain an image of the drop in Step 102. In certain embodiments, that image is used to determine the nozzle radius in Step 103, unless the nozzle radius is known a priori. The nozzle radius R can be determined in the same manner that the minimum thread radius is determined in subsequent steps 109-112. The nozzle radius R is stored in a memory for use in calculating the surface viscosity of the liquid.

[0036] Further liquid is provided to the nozzle in Step 104 so that a thread will form between the nozzle and the drop. An image is acquired of the newly-formed thread in Step 106, and the initial minimum thread radius is determined from e image. The initial minimum thread radius h.sub.0 can be obtained in the manner identified in Steps 109-112. The initial minimum thread radius is stored in the memory for use in calculating the surface viscosity.

[0037] The sequence of Steps 107-114 is repeated at sequential time intervals as the thread increases in length and eventually pinches off. Thus, liquid is provided to the nozzle in Step 107 in a controlled manner to avoid any perturbations in the growth of the thread that might result in bad data. An image of the thread at the current time interval is obtained in Step 108 and this image is digitized in Step 109. It can be appreciated that the image of the thread can be a digital image or can be subsequently digitized. The edges of the thread along the entire length of the thread, and optionally the edges of the drop, are detected from the digitized image in step 110. In Step 111, the minimum dimension between the edges along the length of the thread are located and this minimum dimension is measured in Step 112. In Step 113, the dimension and current time interval is stored in the memory for subsequent use in generating the h.sub.min vs. τ line. In the conditional Step 114, if the minimum dimension measures in Step 112 is zero, or at least within an error range of zero, no further images are obtained because the thread has pinched off. However, if the measured minimum distance is non-zero, control returns in loop 119 to Step 107 to acquire and analyze a new image of the thread as it continues to grow. It is understood that the flow of liquid to the nozzle can be at a continuous uniform rate, so that Steps 104 and 107 of providing “further liquid” happen continuously throughout the measurement process. The images are acquired during the continuous growth of the thread, with images acquired at predetermined time intervals calibrated to provide a sufficient number of images to be digitized between thread formation and thread pinch-off.

[0038] Once pinch-off of the thread has been detected, the method turns to finding a value for the surface viscosity. In Step 115, the stored h.sub.min vs. τ is used to determine a linear relationship. In Step 116, the slope of that line is determined, and in Step 117 the surface tension of the liquid is calculated from the value of the slope, using Equations (1) or (2) described above. The result at the end 118 of the measurement method disclosed herein is a value for the surface viscosity of the liquid that can be used in other processes to more accurately control drop formation and/or more accurately determine drop size, spray patterns, and the like, depending upon the particular application.

[0039] Although the illustrated embodiment relies on drop formation to measure surface viscosity, the methods described herein can be used with other techniques for forming a liquid thread. For instance, one instrument relies on a liquid bridge between parallel rods in which the rods are gradually moved apart to “stretch” the liquid bridge until it pinches off or breaks. With this instrument, the method for measuring surface viscosity follows the steps of FIG. 9. In the first Step 201, a liquid drop is introduced between the ends of two rods, which is equivalent to Step 101 of supplying liquid to the nozzle to form a drop. The ends of the rods are moved apart in Step 202 to form a thread, which is equivalent to Step 104 of the flowchart of FIG. 8. An image of the thread is acquired in Step 203 and initial minimum thread radius is determined in Step 204 from that image. The rods are moved further apart at a constant rate in Step 205, with this movement continuing through the ensuing steps of the method in FIG. 9. Steps 206-211 and loop 212 parallel Steps 107-114 and loop 119. Once the minimum dimension has been found to be zero in the conditional Step 211, the Steps 115-118 are conducted with the minimum dimension data generated in Steps 206-211. In particular, the same equations (1) and (2) can be applied to determine the surface viscosity of the liquid even using the liquid bridge approach. It is contemplated that similar adjustments can be made to the method of FIG. 8 to accommodate other techniques for forming a liquid thread and expanding the thread until pinch-off.

[0040] Those skilled in the art will recognize that numerous modifications can be made to the specific implementations and embodiments described herein. The implementations should not be limited to the particular limitations described, as other implementations may be possible within the scope of this disclosure. For instance, in the illustrated embodiment, a camera and image processor are configured to determine the change in minimum thread radius h.sub.min from photographic images of the drop and thread taken over the life of the thread. Other means may be implemented to determine the minimum radius of the thread, such a system for measuring the change in electrical resistance of the thread as the thread radius decreases over time. In a system of this type, a baseline measurement at a fixed, known radius (or diameter) as necessary to calibrate the relationship between electrical resistance and the minimum radius (or diameter) of the thread