Indian Journal of Fi bre & Texti le Researc h Vol. 28, June 2003, pp. 1 23-133

Fiber migration theory of ring-spun yarns

Mishu I Zeidman

Shenker College of Textile Tec hnology, Ramat-G an, Israel

• Paul S Sawhney"

Southern Regional Researc h Center, Agric ultural Researc h Servic e, USDA, New Orleans, LA 70 1 24, USA

and

Paul D Herrington

Department of Mec hanic al Engi neering, Universi ty of New Orleans, New Orleans, LA 70 148, USA

Received 30 September 2002; revised received and accepted 13 February 2003

The well-known Treloar-Hearlc (T-H) theory of fi ber migration, although more reali stic i n defini ng the yarn struc ture than the helic al model, suffers from certain i nternal i nconsi stenc ies suc h as si ngularities at the yarn c ore. This paper aims at re-exami ni ng the theory of fiber migration and establishing a new basis F or the development of an improved model of ring­spun yarn struc ture wi th an emphasi s on revi sit ing the fiber migration theory. Acc ordingly, the exi st ing theories of Treloar and Hearle et ai. have been reformulated and combi ned in a way that is more acc urate for predic ti ng the struc tural dynamic s of a yarn. Thi s gi ves a more acc eptable desc ri ption of yarn struc ture and, hence, leads to more acc urate predic ti ve models of load-deformation behavior of the yarn. A relati onship between the proc ess parameters and the yarn struc ture is also suggested, whic h is i mportant for the resulti ng yarn properties suc h as tensile strength, abrasion resistance and twist torque. Although it is realized that fiber migration and yarn mec hanic s are complex phenomena, this study is expec ted to contri bute to the analysis and fundamental understandi ng of the phenomena.

Keywords: B ic omponent yarn, Fiber, Fiber migrati on models, Ring spinning

1 Introduction The foundations of structural analysis and

mechanics of ring spun/twisted yarn have been laid by early investigators on the assumption of axial uniformity of yarn structure l . 16. Most of these investigators developed models for load-deformation behavior of yarns, based on the so-called helical or cylindrical models of yarn structure. However, it has always been obvious that a helical structure is theoretically deficient in continuous fi lament and staple fiber yarns. In this context, the pioneering observations of staple yarn structure by Morton and Yen17, Morton lX, Ridingl9• 2o and others suggested the need for a different description of yarn structure, ring­spun or otherwise, to understand a yarn's mechanical behavior. In 1 965, Treloar2 1 and Hearle et al?2-26

proposed models of migration that are now often called migration theory.

"To whom all the correspondenc e should be addressed. Phone: 2864568; Fax: 00 1 -0504-28644 19 E-mail : apsi ngh@srrc .ars.usda.gov

The Treloar-Hearle (T-H) model, although more realistic in defining the yarn structure than the helical model, suffers from certain internal inconsistencies such as singularities at the yarn core2 1 • 22, 26. Therefore, this study was aimed at re-examining the theory of fiber migration and establishing a new basis for the development of an improved model of ring-spun yarn structure with an emphasis on revisiting the fiber migration theory. In summary, the existing theories of Treloar and Hearle et al. have been reformulated and combined in a way that is more accurate for predicting the structural dynamics of a yarn. This gives a more acceptable description of yarn structure and, hence, leads to more accurate predictive models of load-deformation behavior of the yarn. A relationship between the process parameters and the yarn structure is also suggested, which is important for the resulting yarn properties such as tensile strength, abrasion resistance and twist torque.

Most notably, in the context of structure, Koechlin9

seems to be the first to introduce the concept of twist multiplier in 1 829 and Mliller27 to introduce the

1 24 INDIAN J. FIBRE TEXT. RES., JUNE 2003

concept of critical twist in 1880. In the context of mechanics, Gegauff' used the twist multiplier (coefficient of twist) to predict the strength of the yarn from the strength of the component fibers and provided the starting point for mechanical models of yarns developed since 1 907 . In 1 926, PeirceIO

recognized the importance of' these analyses and applied them in the early development of structural mechanics of yarns.

Subsequently, Braschler4 in 1 935, Platt I I in 1 950, Hearle et al.7.8 in '1 958 and 1 96 1 , Treloarl6 in 1 963, Cheng et also in 1 974 and many others developed models for load-deformation behavior of yarns, based on the so cal led helical or cylindrical model of yarn structure. These models spawned the field of structural mechanics of yarns as we know it today. It .is important to emphasize that all these theoretical achievements were prompted by very practical goals, namely the prediction of' mechanical properties of yarns, especial ly the stiffness and tensile strength, in terms of load-elongation behavior of the constituent fibers. Other features of practical interest, such as contraction factor by Braschler.J, internal lateral pressure by Batra et al? 3, bending behavior by Backerl and Platt et all2. , and torsional behavior by Platt et al.13, Postle et a1.14, Thwaites l5, etc . , were also analyzed using the helical model . The helical model gained acceptance because the theoretical results for some mechanical properties based on the helical model were found to be in reasonable agreement with practice. It was, however, obvious that a helical structure is theoretical ly deficient in continuous filament and staple fiber yarns. In the continuous filament case, the model implies differential change in the initial length; in the staple case, additional ly , the free fibers of finite length would not contribute to development of transverse forces across the yarn to give it cohesion and strength.

Despite several additional publications since then, the structure of migration theory, as it is often cal led, remains where it was in 1 965.

This paper re-examines the theory of migration for several reasons. First, the textiles, inc luding those made from twisted yarn structures, are increasingly used as load bearing structural members. The economics of specific strength of these structures demands efficient structural design, which, in turn, requires accurate prediction of yarn load-deformation properties. However, the current models offer only approximate predictions and require relatively large safety factors. Second, to overcome the above

difficulty by using the current migrational models to define the structure requires that the models be internally consistent and devoid of difficulties that restrict the use of discrete-continuum calculus. As both the Treloar and Hearle models have limitations, it is reasonable to ask whether other models can be developed which would eliminate these difficulties, give more acceptable description of structure and hence lead to more accurate predicti ve models of load-deformation behavior.

In the same vein , Sawhney et al. 28.35 have attempted to produce what might be cal led skin-core bi-constituent yarns in which the core consists of man-made fibers (e.g. high tenacity polyester staple or continuous filament) surrounded by cotton such that the yarn is not only stronger than the comparable size cotton yarn, but also has all the surface characteristics of a cotton yarn . To understand the micromechanics of' yarns produced by such new systems, a more detailed understanding of the fiber migration phenomenon is first desirable.

2 Methods 2.1 Fiber Configuration in Ring-SpunlRing-Twistcd Yarns:

Relationship to Process Kinematics

In ring spinning/twisting, an untwisted strand of quasi-paral lel fibers issuing from the delivery rol ls is transformed into a twisted structure (yarn), with nearly circular cross-section, in which fiber configurations are similar to helices about a common axis. The transformation requires time, albeit very short, and spatial distance called the twist triangle.

I t is critical to note that the transformation (twisting) of a parallel fiber bundle into a yarn takes place in the twist triangle only; the process involves displacement of fibers relative to each other and relative to yarn axis. In particular, during a short time dt a length ds of every fiber enters the twist triangle. During the same period, every fiber at [he base of the triangle advances towards the twisted state by a distance dz along the yarn axis and rotates about this

axis through an angle dq;; each fiber element either moves towards or away from the yarn axis a distance

dr. Thus, while ds, dz and drp are the same for al l fibers, the distance dr differs from point to point along each fiber axis and from fiber to fiber at the same cross-section along yarn axis. For a given delivery speed, spinning tension and torque, the constancy of ds, dz and drp are relatively assured by the process, whereas the constancy of dr is not. The latter is governed by the compactness of the twist

ZEIDMAN et ai.: FIBER MIG RATION THEORY OF RING-SPUN YARNS 1 25

triangle, fiber length and its distribution in the fiber stock.

Alternatively, ds equals the arc length on the delivery rollers surface (because of the contact), dz equals the length of yarn wound on the bobbin after

twist contraction, and drp equals the rotational displacement of the twist i nserting element (traveller) . At the same time, the fiber axis i n a yarn fol lows a spatial curve best described by cylindrical coordinates

r, rp and z; the yarn axis is assumed straight and coincident with z-axis. An infinitesimal length of the fiber axis, ds, then constitutes a diagonal of an

elemental parallelepiped of sides dz, rdrp and dr, yielding:

... (1 )

Thus, the kinematics of the twisting process forms the basis for derivation of geometrical relationships descriptive of the yarn structure. In this context, the fiber axis configuration can be regarded as the trajectory of a point moving about, along and against the yarn axis during formation of the yarn. The motion of this point , therefore, can be related to process kinematics as fol lows. Dividing Eq. ( 1 ) by the square of the corresponding time interval dt yields:

. . . (2)

where v,,:= ds is the point velocity along the fiber dt

. dz h . I ' I h

. aXIs; Vw:= - , t e POlOt ve oClty a ong t e yarn aXIs;

dt

to:= d<Jl, the angular velocity of the point about the dt

. d dr h

. . I ' N . yarn aXIs; an VIII := dt' t e migratIOn ve oClty. ote

that the velocities defined thus relate to a point moving along the fiber axis. The process of yarn formation, however, suggests that these enti ties must be related to the k inematics of yarn formation, as mentioned above, assuming no fiber slippage. Thus, v" is the surface speed of the delivery rol ls, Vw is the winding velocity of yarn on the bobbin, and w is the angular velocity of the twist insertion element. In a normal ring spinning operation, these velocities are nearly constant and can be controlled independently.

The ratio � = T", is called the machine or calculated V"

twist, the ratio � = T is the real twist in yarn wound VIfJ

on the bobbin, and the ratio � = C is the contraction VIII

factor of the yarn, resulting in the usual relationship T = T",C. Therefore, Eq. (2) is a kinematic analog of the yarn structural geometry. An alternate form of the same fundamental relationship can be obtained as:

(dS]2 =1+r2(d<Jl]2 +(dr]2 dz elz dz

. . . (3)

The local twist rate or twist frequency T(z) is defined as:

T(Z)= (d<Jl] elz _

... (4)

and is a geometric characteristic of the fiber path in yarn. To equate it with the usual twist in the yarn (T), which is a kinematic· characteristic of the twist insertion process, one assumes that torsional deformation of the yarn is: (i) the same for all fibers i n the strand cross-section, and (i i) constant along length of the yarn. Of course, i t is implicit that the overall fiber characteristics are uniform and the yarn has a constant number of fibers in every cross-section.

The twist period or pitch (h = 27dT) i s the yarn length corresponding to one turn of twist. r is called the twist angle and is defined as:

1: := tan -I (�; 1'] = tan -I (Tr) . . . . (5)

Next, the local migration rate m(z) and the

migration angle f.1. are defined as:

m(z)=l' dr] dz ,

!! := tan -I ( �: ] = tan -I (m) .

... (6)

. . . (7 )

In addition, the fiber inclination angle (relative to yarn axis) or the fiber angle e, for short, i s defined by:

e _1(dS] := sec - . . dz

. . . (8)

By virtue of Eq (I), the tangent of the fi ber angle can be expressed in terms of the twist and migration angles as:

. . . (9)

or in terms of the structural parameters Tand mas:

... (10)

This form of the fundamental equation of yarn geometry, derived by Zeidman, shows the two key features of yarn structure, namely twist and migration,

1 26 INDIAN J. FIBRE TEXT. RES. , J UNE 2003

to be inseparable manifestations of the same twist insertion phenomenon.

In general, both T and m are variable functions of z

describing fiber axis trajectory of a general configuration. However, the winding velocity and angular velocity of the yarn during ring spinning/winding maintain their respective sense, and these characteristics are passed along to every

constituent fiber. Therefore, qXz) is a monotonical ly increasing function and the fiber path is a helical curve around the yarn axis. Furthermore, since in ring spinning the ratio of spindle rotational speed to winding speed is nearly constant, the twist T is also constant. As a result, the curve described by the fiber axis around the yarn axis may be considered a helix of constant pitch but variable radius (m of; 0).

2.2 Development of Fiber Migration Model

Complete migration implies that the fiber migrates from yarn surface to yarn core in one half cycle (of migration) and then back to the surface in the other half. I ncomplete migration means that within the limits of a single migration cycle the fiber migrates from the surface inwards but is turned around to migrate outward before reaching the yarn axis. Complete and incomplete migration cycles are both present along the same fiber. The model that fol lows is developed for complete migration cycles only.

The fol lowing represents a set of hypotheses relative to migration of fibers, necessary in building a model of complete migration . The hypotheses considered here are directly relevant to yarn structure and its mechanical behavior.

I . The fiber configuration does not admit kinks, implying that migration rate is a continuous function.

2. Within the l imits of a semi-cycle of migration, the helix radius is a monotonic function of its

argument z or rp. 3. Migration in its complete cycles is uniform, both

along each fiber and between fibers. Therefore, the parameters and laws of complete migration are the same throughout the yarn structure.

4. The density of packing of fibers in the yarn is uniform throughout the yarn volume, implying a constant packing factor. Local measures of packing, relating to small yarn volumes, although not constant, change very little in regular, compact yarns.

2.3 Proposed Model of Complete Migration

Experience dictates that migrating fibers at the yarn surface, in general, do not have kinks, the helix

passing smoothly from increasing to decreasing radius with no discontinuity. Therefore, by hypothesis 1 , the migration rate of the fibers at the surface is zero or that

meR) =0 . . . ( II) where R is the outer radius of the yarn. Thus, we introduce Eq ( I I ) as a boundary condition.

I n developing the model for complete migration, we use Treloar's definition of packing density (length of fiber per unit volume of yarn) but determine it in a different way. When appl ied to finite yarn segments of length Z and cross-sectional area (assumed y constant) A , the yarn volume V equals the product y y AyZy. I f the length of the fiber in this volume is LJ, the

overall packing density qv is the ratio

... (12)

The subscript v attached to q is to specify that it relates to the yarn volume. Now, let us consider only the area of a cross-section of the yarn, Ay- The fibers

intersecting a cross-section are seen as mere points; let Nr be the number of these points, assumed

constant. We can now define a different packing density q" by dividing Nr by AI':

_ NJ q"=A y

... ( 13)

Let us call this new measure as number density (N­density), representing the number of fibers intersecting a unit area of the yarn cross-section. The N-density is a sectional measure. It can be related to the volumetric packing density as fol lows: each point on the yarn cross-section is a result of cutting the yarn. Multiplying Nt by the average length of each

fiber in the yarn segment, SJ' we obtain the total fiber

length Lf' Therefore,

LJ NJSJ SJ ql'= AT= AT=q"z y !' Y J J

... (14)

where Z( is the length of a finite fiber segment. I n other words, the packing density and N-density are directly proportional to each other, the proportionality factor being equal to the contraction factor. Determining q" and q" requires experimental

knowledge of the cross-sectional areas of the fiber and yarn . Thus, if Ar is the average fiber cross-

ZEIDMAN et at.: FIBER MIGRATION THEORY OF RING-SPUN YARNS 1 27

sectional area and VJ is the average fiber volume, then:

v LA c==

+= 2.....1...=qA

V V • J y y

. . . (15)

where C, the contraction factor, can be i ndependently determined, for example as the ratio of the length of the untwisted strand to that of the final twisted yarn. As for q", it is simply the ratio of the number of

filaments N (fibers in the cross-section) to the yarn cross-sectional area Ay:

... ( 16)

where R is the yarn cross-section radius. To apply the two measures of density of packing to

small volumes, let us consider an elemental annulus about the yarn axis, of radius r, width dr and height dz. Its volume dV" is:

dVy = 2

rrrdrdz . . . . (17) Let dn be the number of fibers coming from the

yarn cross-section that intersect the annulus and ds the length of each fiber in the annulus. Then, the length of fiber in the annulus (dLf) i s:

... (18)

Now, we can define a local volumetric packing density q.,(r) as the ratio of dLfto dV,,:

dLr dn ds q(r}=-' = -- .- .

" dV, 2m·dr dz . . . (19)

The reason for separating qJr) into two factors is

obvious: the first factor represents the number of fibers intersecting the cross-section of the annulus divided by its area. I t is therefore a local N-density measure, q,,(r). Then, qJr) becomes:

ds qJr} = qJr} dz

. . . . (20) Dividing qJr) by qJr) we obtain a normalized

packing density per) that can be used to characterize the packing of fibers in yarn:

() q,,( r) pr = --. q,.( r)

... (21)

Notice that p( r) is dimensionless and does not have the disadvantage of being infinite close to the yarn axis,

because both qJr) and qn(r) tend to infinity when r

tends to zero, but their ratio does not necessarily do so. Next, we argue that while qJr) and qJr) may vary

with radius, both qJr) and qJr) increase as the radius

decreases; changes i n both are mainly determined by lateral inter-fiber forces developed in the structure due to i mposed twist and spinning tension. Therefore, it is reasonable to postulate that their ratio remains essentially constant across the yarn, i .e .

p( r) = p = constant . .. . (22) By Eq ( 1 7) , this ratio is equal to (ds)/(dz), which, in

turn, equals the secant of the fiber angle:

ds -== sec e = constant . dz

. . . (23) Constancy of the fiber angle is a principal result of

the new hypothesis and constitutes an i mportant feature of the new model. In order to build a model based on this assumption, let us first determine the function m( r) using Eq ( 10):

J 2 " 2 lI1(r)=± tan 8-T-,. . ,

.. (24) We can now use the hypothesis of zero migration rate at the yarn surface [Eq ( 1 1 )] to obtain:

tan e = TR .,

.(25) showing how easily tpis basic parameter can be determined from the usual structural parameters T and R. Therefore, m(r) can be expressed as:

m( r) = ± rJ R2 -,.

2 . ., .(26)

The migration rate at the yarn axis m(O) i s then:

1110 == m(O) = ± TR. . .. (27) The minus signs in Eqs (26) and (27) are valid for

the first half of the migration cycle and the plus signs are valid for the second half.

Now we can determine the helix configuration described by the function r(z). Since mer) is the first derivative of r with respect to z, and m(r) == (dr)/(dz), we need to solve the fol lowing s imple differential equation:

dr ±Tdz ... (28)

whose solution is

sin-I � = ± T z + b . R

. ..

(29)

1 28 INDIAN J. FIBRE TEXT. RES., JUNE 2003

To determine the constant of integration b, we consider a cycle of migration starting at the yam surface. The corresponding boundary condition r = R at z = 0 yields:

b =

2: 3n (2n -I)n 2' 2 '"'' 2

and Eq (29) becomes:

• _I r ( 21l-I)n • Sill -= ±7z R 2

... (30)

... (31)

where n is an arbitrary integer. The radius r can now be expressed explicitly as:

r=Rsin [( 211;1)1l'

±T,l ... (3 2)

From trigonometry we know that sin (n ± x) = - sin

d . n x, an Sin (2' ± x) = cos x. Therefore,

, = Rlcos (Tz)1 . ... (33) The fiber considered In our case has a complete

migration, which requires r = 0 at z = H. This condition yields:

0= cos(TH)

which is satisfied if

H=� � 2T ' 2T

( 21l-1)n 2T

... (3 4)

... (3 5)

At this point, we must use the hypothesis 2, denoting monotonic migration behavior. This condition is only satisfied if n = I in Eq (35) .

Therefore, Eq (33) is valid provided that 0 < z S; �, 2T

i.e. in the first semi-cycle of migration . Thus, the above complete migration model captures

the periodic character of the migration rate, well documented in practice. The derived equations for m and r require knowledge of only two yam features, namely twist and radius. The simplicity of this is quite unexpected, at least at this stage of our development. Since no characteristic of the yam formation process has been taken into consideration, the veridicity of the model should be questionable. In addition, there is an experimental result that contradicts the above model, namely the migration period that is characterized by H. Using the twist period (pitch) h = 2n1T and substituting into Eq (35), we obtain for the model :

H= !.!.... 4 ... (36)

In practice, however, the ratio of migration period to twist period is found to be much higher and to differ from yarn to yarn. To overcome this discrepancy between the results obtained from the proposed model and experiments, the migration rate must be lower, to be consistent with the length of the migration period. A possible solution to finding such a correction is to multiply m( r) in Eq (26) by a constant factor, such as a. Such a solution would keep the simplicity of the model and, more importantly, the periodic character of migration .

Introducing the factor a , Eq (26) becomes:

lIl(r)=±a� such that

IIl(R) = 0 and m(O) = aTR .

... (37)

...( 38)

I f the factor a is considered constant, then the new equation for r(z), obtained by integration, becomes:

r( z) = R Icos(aTz)1 . ... (3 9)

To determine the factor a, we use the boundary condition:

r(H) = Rcos(aTH) = 0 ... (4 0)

which is satisfied within the l imits of the first semi­cycle of migration if

1t aTH=2 or a = n

2TH .. . (41)

With this substitution, the migration rate m(r) becomes:

m(r) = ± ...!!:...-� R2 - ,2 2H

... (4

2) Eq (42) is a primary equation of the new model. Applying it requires, however, the k nowledge of an additional magnitude, viz. the parameter H, the length of half of the migration period. Notice that the Treloar-Hearle model also requires the knowledge of H in addition to several other structural features of the yarn (number of fibers per cross-section and packing density) that are not necessary in the new model . In practice, it was found that the migration period is directly proportional to the twist period36, the proportionality constant factor ( g) and the ratio of the migration period to the twist period (pitch) h. It is dependent on factors such as the number of fibers in the yarn, yarn radius, elastic properties of the fiber, etc. The migration factor g seems to vary l ittle with other structural yarn features and therefore:

ZEIDMAN el £II.: FIBER M IGRATION THEORY OF RING-SPUN YARNS 1 29

2H 2H TH g = h

= 21t = -;-

= 2a

T

. . . (43)

By Eqs (4 1 ) and (42), r(z) can be written in terms of T and gas :

T r(z) = R leos(2g z)1

or in terms of H, as shown below and in Fig. l :

7f r(z) = R leos(2H z)1 .

. . . (44)

. . . (45)

In cylindrical coordinates, the helix configuration can be completely described by the two functions: r(z)

and r( rp). If twist i s assumed constant both along each fiber and across the yarn, then z and rp are directly proportional to each other, and examinations of r(z) and r( rp) are equivalent. Nevertheless, some particular features of migration are better revealed when using r(z), other features when using r(rp), and sti l l other features when using both.

In the newly derived model, the helix radius can be written as a function of the rotation angle rp as:

7f r( rp} = R leos(2H.T rp)1 = R leos(arp)l· . . . (46)

This is also the equation of the spiral representing the cross-sectional projection of the fiber helix. The equations for r(z) and r( rp) have been derived for the first half of the migration cycle, except for m( r), which is valid for the whole cycle. The equations for the second half of the cycle are the same as for the first cycle since sin(1'[ - B) = sinS, and Icos(1'[ - (3)1 = IcosBI·

3 Results and Discussion Introducing the parameter a (or g or H) into the

preliminary model, to better fit to actual migration

IX: i:: 0.8 II> :::l :0 0.6 .. 0::

"t:I .� 0.4 n; E o 0.2

Z

o+------.------. .-____ -, ______ �----� o 0.2 0.4 0.6 0.6

Normalized Yarn Length (zlH ) Fig. 1-- Migration radius vs axial length

periods, represents a correction that alters some basic features of the model. The most affected is the fiber angle 8. Substituting the modeled migration rate mer) from Eq (37) into Eq ( 1 0) yields:

tan2S = m2(r) + f2r2 = c?f2R2+ (I _(2)f2r2 . . . (47)

or . . . (48)

Eq (47) emphasizes the influence of the hel ix radius rand Eq (48), the influence of the difference between the maximum and the local radi i . The second term in Eq (48) is the difference between the maximum fiber angle at the yarn surface and the local fiber angle. This difference increases as the fiber approaches the yarn axis . The minimum value at this point ( r = 0) is :

tanSo = aTR . . .. (49)

The preliminary new model, unadjusted, can be obtained as a particular case of the adjusted model by making a = I .

I f we restart building the model from the point where the fiber angle was equal to the number packing density, we must admit that any variation of the fiber angle implies a variation of the normalized packing density . However, a significant deviation in the value of the fiber angle, in the range of moderate angles, results in a small deviation of the packing density . This is because the number packing density equals the secant of the fiber angle which by trigonometry is :

seeS = � I + tan2S . . . (50)

If the range of tan8 is low, for example with fiber angles less than 45°, the i nfluence of the second term under the radical is not critical. For example, if the fiber angle at the yarn surface is 30° and at the yarn axis is 5", the decrease of the number packing density from surface to core is only 1 5%. In other words, the number packing density p increases faster than the volumetric packing density q with a decrease of the radius of only J 5%. Therefore, even if the total relative deviation of the fiber angle in the adjusted model is significant, its influence on the number packing density is rather low. Thus, the acceptance of the adjustment discussed above does not imply a complete reversal of the basic hypotheses of the model .

For easier comparison, aor! for analytical purposes in general, it is important to use non-dimensional measures. The most natural measures for normaliza-

1 30 INDIAN J. FIBRE TEXT. RES., JUNE 2003

tion are the following: outer radius of the yarn (R ), local helix radius (r), migration half-period (H) and the distance along the yarn ( z). In this way, the helix configuration is described in normalized cylindrical

coordinates: p == � , rp and S == �. Using these

substitutions in Eq (44), we obtain the equation of the

normalized helix radius p for the new model as:

7r . A Q = R Icos(2 0 I .... (51)

and from Eq (42) we obtain the migration rate m(p) as

a function of the normalized radius pas:

7r'R_� TR_ � lI1(p) = ± 2H \J I -� = ± 2'M \J I - � . ... (52)

Differentiating IX SJ with respect to (, we obtain an

equivalent migration rate n(p):

related to the regular m through

H n(p) = m(PfR .

... (53)

... (5 4)

When the normalized radius p decreases from 1 to 0, the absolute migration indicator m increases from

zero to !!:Ii and the absolute n increases from zero to 2H

n 2

Using Eq ( 1 ) and considering the case where dz = 0, a sectional projection of a fiber element is given by its length dl and can be determined from:

dl2 = dr2 + r2'dql . . . . (55)

Dividing the above terms by dql- , we obtain:

dl 2 dr 2 , (drp) = (drp) + r- . . . . (56)

The first term in the right hand side, derivative of

the spiral radius with respect to the angle rp , can be called projective migration rate s. It can be related to the usual migration rate m through:

m=sT ... (57)

which results by simple substitution . Therefore, the two migration rates m and S are directly proportional to each other and their use can be interchanged.

As an application we can determine the fiber angle in terms of the projectional migration rate s instead of the usual migration rate m. Substitution yields:

... (5 8)

By the same token we can determine the ascension

angle of the helix (n between the tangent to the fiber

axis and the yarn cross-section. Since 8 + Y= 1T.I2, we have:

2 I tan Y= 2 2 2 . T (r + s )

... (59)

Returning to the spiral projection of the fiber, we can determine the two complementary angles describing its configuration, which are the most significant features of cross-sectional migration: (I) between tangent to the spiral and the local spiral

radius a , and (2) between the same tangent and the

tangent to the local circle concentric with the spiral, j3 = nl2 - a. These can be determined as:

nip r tana = dr = �

and dr lan�=-d r' rp s r

... (60)

... (61)

Applying these results to the model developed gives:

lana h . .. (62) a' 1 -p2

which is shown in Fig. 2. The above equation shows that at the yarn surface, where p = I, the fiber projection is perpendicular to the radius and therefore tangent to the yarn surface. At the yarn axis, where p = 0, the fiber" projection is also perpendicular to the yarn axis. In this case, the spatial fiber angle is less than 7tl2 and, therefore, the fiber itself is not perpendicular to the axis.

3.1 Fiber Migration in Yarns and Spinning Process Parameters - Determining Yarn Contraction Factor

Next, let us associate the new model with the usual process parameters in ring spinning processes. The

6

o C 4 .!

O+-�==�==�=----.-----r----� 0.2 0.4 0.6

Nonnalized Radius ( r/R ) Fig. 2- Tan a vs normalized radius

0.8

ZEIDMAN el l/I. : FII3ER MIGRATION THEORY OF RING-SPUN YARNS 1 3 1

terms describing the configuration of fiber paths in the yarn can be related to the kinematics of yarn formation in the fol lowing way. The speed along the yarn axis, (dzldt) = vp, equals the winding speed of the yarn Vy with negligible alternate deviations, i .e. VI' = V The term (dqidt), the rate of twist insertion, is y associated with and practical ly equal to the angular speed of the twister (j). In ring spinning, the ratio of (j) to V equals the machine twist rate, which is

maintained constant even in transition states. If the diameter of the yarn cross-section is constant, the yarn twist T is also constant along its length and equal to the machine twist. The term (ds/dt) == v is the time J derivative of the fiber length, i .e. the length of fiber entering the yarn structure in a unit t ime (the speed of fiber insertion in the yarn) . In normal operating conditions, this is equal to the surface speed of the delivery rollers (vd) because the fibers at the delivery l ine are assumed parallel to the direction of their motion and no slippage of the fibers relative to the delivery rol ls occurs, except accidentally. Therefore, one may assume that each point along a fiber leaves delivery roll s and enters the yarn structure with constant speed relative to the speed of yarn formation, and that this speed is the same for all fibers. Therefore, VI = vv' In normal operating conditions, the ratio of vd to Vy is practically constant and is defined as the contraction factor of the yarn, C, that is

c '" � = constant . . . . (63) Vy

Using the substitutions suggested by the above equalities, we obtain:

. . . (64) Generally speaking, applying the above equation to

describe fiber behavior (configuration) in a yarn requires that C be considered differentially, for each point along each fiber, and be equated with the ratio (ds/dz)r' But the ds and dz values are measured at two different locations: ds at the delivery rollers, beginning of the twist triangle, and dz at its apex . If both ds and dz are measured at the delivery nip l ine, ds would be the same for al l fibers, while dz would vary from fiber to fiber because the fibers orient themselves in (at) different local orientations (angles relative the yarn axis). Alternatively, we can measure both dz and ds after the yarn is formed. Then dz is the same for all fibers, while ds differs from fiber to fiber.

However, (ds/dz)r in general is not constant if applied locally .

In the preliminary new model , (ds/dz)r is constant. If we substitute r = R and m(R) = 0, we obtain :

. . . (65)

Eq (65) emphasizes one of the main relationships between the spinning process/product represented by C and the yarn structure represented by 8.

In the new model adapted to real migration periods, (ds/dz) = sec 8 varies within a certain range from fiber to fiber in the same yarn cross-section or, equivalently, along the same fiber in a complete migration cycle. In this case, the contraction factor can be determined as an overall (average) value along the fiber as:

. . . (66)

where Sc is the fiber contained in length H of yarn. To determine Sc , we start from Eq ( l ) describing

the fiber configuration in yarn and substitute m(z) and r(z) from Eq (45) . Thus,

ds= 1+ 1t2 .R: sin2 (�.zJ + r2 .R2 cos2 (�.ZJ dZ . . . . (67)

4.H 2.H 2.H

Eq (67) can also be written in the fol lowing form:

ds = 1 + r .R -r .R (1- a ).SIn -.? dz. 2 2 2 2 2 · 2 ( 1t J 2.H . . . (68)

to obtain a (Legendre's) normal e l l iptic integral of the second kind. Using the substitutions :

7r X =

2.H z , 2·H dz =� dx , and e - r2R 2 (I _ a 2 ) - l + r 2R2 '

. . . . (69)

the above equation can be brought to a regular normal form as :

ds = 2H --.J J + r2·R2 . �I - k2 sin 2 x dx. 7r . . . (70)

Integrating for the first semi-cycle of migration (x = 0 . . . 1t/2, when z = 0 . . . H), yields:

2H rrI2 r----

Sc = -� J + r2·R2 .J � J - k2·sin2 x dx . 7r 0

. . . (7 1 )

Then, the contraction factor can be determined as:

c = � = �--.JJ + r2·R2 £(k,7tl2) . . . (72)

1 32 INDIAN J. FIBRE TEXT. RES .. JUNE 2003

where E(k;rr/2) is Legendre' s complete normal e l l iptic integral, defined as:

nI2 E(k,nI2) = f -J J - k2 ·sin2 x dx

o

4 Conclusions

. . . (73)

The Treloar-Hearle model or theory of fiber migration in yarn is based on a set of seemingly plausible hypotheses that capture several important features of the ring spun/twisted yarn structure. Yet, the model leads to certain inherent difficulties in the conclusions that the yarn core must be hollow and the fibers must develop kinks on the yarn surface, which is not the case in case of the truly co-axial core-wrap yarns produced by the USDA-patented staple- core ring spinning technology. Based on the fiber migration model presented in this paper, some fibers in a ring-spun yarn structure migrate from the yarn surface to the yarn core in one half cycle and then back to the surface in the other half, while most fibers migrate from the surface inwards and then turn around to migrate outward before reaching the yarn axis or core. The degree or extent of fiber migration in each scenario determines the ultimate yarn structure, which, in turn and in part, determines some of the yarn characteristics and, hence, qual ity. The old models of fiber migration resulted in a yarn structure with a hollow core where no fibers could enter, which does not explain the formation or structure of a solid, all staple, bi-component yarn spun on the ri ng spinning system. The relationship between the spinning process parameters and the yarn structure and properties is emphasized.

Acknowledgement This study was an outgrowth of the investigation of

the structure of sheath-core bi-constituent yarn of the type reported by Sawhney et.al,. The study was conducted under a research project sponsored by USDA, ARS, SRRC. The abi l i ty to produce such a yarn is undoubtedly due to the control of migration of fibers in the two constituents (core and sheath) of the yarn . While development of the models reported in this paper d id not emerge until after termination of the ARS-SRRC sponsored project, the motivation i t provided cannot be denied.

The authors are greatly indebted to Drs S. K. Batra and M.W. Suh of College of Texti les, NCSU, Raleigh, NC, and Dr K . Q. Robert of SRRC for their significant input in the in itial work. In addition, they

are also indebted to the state of NC for indirect support it provided through its support of research in the College of Texti les, NCSU. They are also grateful to Prof. H. D. Wagner of the Weizman Institute, Israel, for permitting the use of the institute 's research faci l i ties.

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